technische universit¨at berlin...matrix invariants and semi-invariants we are also able to give...

Technische Universitat Berlin

Institut fur Mathematik

Fakultat IIStr. des 17. Juni 136

10623 Berlinhttp://www.math.tu-berlin.de

Master’s Thesis

Invariants and Orbit Closure Problems for

Quivers and Generalizations to Tensors

Mahmut Levent Dogan

Matriculation Number: 395900November 26, 2019

Supervised byProf. Dr. Peter Burgisser

Second readerDr. Mario Kummer

http://www.math.tu-berlin.de

Hereby I declare that I wrote this thesis myself with the help of no more than thementioned literature and auxiliary means.

Berlin, November 26, 2019

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(Signature [Mahmut Levent Dogan])

Hiermit erklare ich, dass ich diese Arbeit selbst mit Hilfe der genannten Literaturund Hilfsmittel verfasst habe.


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(Unterschrift [Mahmut Levent Dogan])

Bu tezi, belirtilen kaynaklar ve yardımcı araclardan baska bir sey kullanmadankendi basıma yazdıgımı beyan ederim.


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(Imza [Mahmut Levent Dogan])

Abstract

In this master’s thesis, we study the invariant theory of three group actions over alge-braically closed fields of arbitrary characteristics.

Firstly, we investigate the invariants of matrix tuples under simultaneous conjugationand the left-right action. In the literature, these are called matrix invariants and matrixsemi-invariants respectively. In both cases, we give a set of generators for the ring ofinvariants and introduce concrete degree bounds for the degree of the generators. Wealso give efficient algorithms for orbit closure intersection problems of both simultaneousconjugation and the left-right action.

Secondly, we analyze invariants and semi-invariants of quiver representations underthe action of a product of general linear groups. We give a set of generators for theinvariant ring using ideas from the matrix case. We also give a set of generators for thesemi-invariants of quiver representations under the assumption that the quiver containsno oriented cycle. In this case, we also give a description of the nullcone of quiverrepresentations under the action of a product of special linear groups. As in the case ofmatrix invariants and semi-invariants we are also able to give some degree bounds forthe generators of the ring of invariants and semi-invariants of quivers.

Lastly, we study order three tensors and the corresponding nullcone using the ideasfrom previous sections. We introduce an equivariant map from order three tensors toforms. We introduce an important invariant of order three tensors called Cayley’s hyper-determinant and describe its relation with the discriminant of forms. Moreover, usingthe mentioned equivariant map, we are able to give a description of the nullcone of orderthree tensors in the cases of two slices tensors and cuboids.

Zusammenfassung

In dieser Masterarbeit untersuchen wir die Invariantentheorie von drei Gruppenopera-tionen uber algebraisch geschlossene Korpern beliebiger Charakteristiken.

Zunachst untersuchen wir die Invarianten von Matrixtupeln bei gleichzeitiger Kon-jugation und der Links-Rechts-Operation. In der Literatur werden diese Matrixinvari-anten bzw. Matrixhalbinvarianten genannt. In beiden Fallen geben wir eine Reihevon Generatoren fur den Ring der Invarianten an und fuhren konkrete Grenzen fur denGrad der Generatoren ein. Wir geben auch effiziente Algorithmen fur Orbit-Closure-Schnittprobleme sowohl bei gleichzeitiger Konjugation als auch bei der Links-Rechts-Operation an.

Zweitens analysieren wir Invarianten und Halbinvarianten von Darstellungen von Ko-chern unter der Operation eines Produktes allgemeiner linearer Gruppen. Weiterhingeben wir eine Reihe von Generatoren fur den invarianten Ring unter Verwendung vonIdeen aus dem oben genannten Fall an. Wir geben auch eine Reihe von Generatoren furdie Halbinvarianten von Kocherdarstellungen unter der Annahme an, dass der Kocherkeinen orientierten Zyklus enthalt. In diesem Fall geben wir auch eine Beschreibung derNullfaser von Kocherdarstellungen unter der Operation eines Produkts spezieller linearerGruppen an. Wie im Fall von Matrixinvarianten und Halbinvarianten konnen wir auchfur die Generatoren des Ringes von Invarianten und Halbinvarianten von Kochern einigeGrenzen fur den Grad der Polynome angeben.

Zuletzt untersuchen wir die Tensoren dritter Stufe und die entsprechenden Nullfasernunter Verwendung der Ideen aus den vorhergehenden Abschnitten. Wir fuhren eineaquivariante Abbildung von Tensoren dritter Stufe zu Formen ein. Weiterhin fuhren wireine wichtige Invariante dieser Tensoren namens Cayleys Hyperdeterminante ein undbeschreiben ihre Beziehung zur Diskriminante von Formen. Daruber hinaus konnenwir unter Verwendung der erwahnten aquivarianten Abbildung eine Beschreibung derNullfaser von Tensoren dritter Stufe fur Falle von Tensoren mit zwei Schichten oderCuboids geben.

Oz

Bu yuksek lisans tezinde, cebirsel olarak kapalı herhangi bir karakteristikteki cisimlertemelinde uc grup etkisini inceliyoruz.

Ilk olarak, matris sıralılarının es-zamanlı eslenik alma ve sag-sol etkisi altındaki degis-mezlerini arastırıyoruz. Literaturde bunlara sırasıyla matris degismezleri ve yarı-degis-mezleri deniyor. Her iki durumda da, degismezler halkası icin bir uretec kumesi veriyor veureteclerin derecesi icin somut sınırlar getiriyoruz. Ayrıca, hem es-zamanlı eslenik almahem de sag-sol etkisinin yorunge kapanıs kesisimleri problemi icin verimli algoritmalarveriyoruz.

Ikinci olarak, genel dogrusal grupların bir carpımının etkisi altında sadak temsillerinindegismezlerini ve yarı-degismezlerini inceliyoruz. Matris durumundaki fikirleri kulla-narak degismezler halkası icin bir uretec kumesi veriyoruz. Ayrıca, sadagın yonlu bircevrim icermedigi varsayımı altında yarı-degismezler halkası icin bir uretec kumesi veriy-oruz. Bu durumda, ozel dogrusal grupların bir carpımı altında sadak temsillerinin bos-koniginin bir tasvirini veriyoruz. Matrix degismezleri ve yarı-degismezleri durumundaoldugu gibi, sadak temsillerinin degismezler ve yarı-degismezler halkasının ureteclerinindereceleri icin bazı sınırlar veriyoruz.

Son olarak, onceki bolumlerden fikirleri kullanarak uc dereceli tensorleri ve karsılık ge-len bos-konigi calısıyoruz. Uc dereceli tensorlerden formlara giden bir es-dereceli eslemeyitanıtıyoruz. uc dereceli tensorlerin onemli bir degismezi olan Cayley’in hiperdetermi-nantını tanıtıyor ve formların diskriminantı ile olan iliskisini acıklıyoruz. Ek olarak,belirtilen es-dereceli eslemeyi kullanarak, iki dilimli tensorler ve kuboidler durumundauc dereceli tensorlerin bos-koniginin bir tasvirini veriyoruz.

Acknowledgements

First of all, I would like to thank my supervisor Prof. Dr. Peter Burgisser for introducingme to the beautiful subject that is invariant theory and giving me the chance to writea master’s thesis under his supervision. He guided me along the process of writingthis thesis and made valuable suggestions. Without his support, this thesis would lackcontent and structure. I also want to thank Dr. Mario Kummer, Dr. Visu Makam, Prof.Dr. Joseph Landsberg, Prof. Dr. Bernd Sturmfels, Dr. Alperen Ergur, Dr. KathlenKohn and Philipp Reichenbach. They all helped me finish this master’s thesis one wayor another. Special thanks go to Dr. Josue Tonelli Cueto: He always had an open doorwhenever I had a question.

I also want to thank everyone who supported me during my Phase 1 studies in BerlinMathematical School. I would like to start with my mother, my father and my sister.They never stopped believing me and supported me in my achievements and in myfailures. I am who I am thanks to them. I also had a great friends in Berlin MathematicalSchool. I especially want to mention Harikrishnan Mulackal, Marco Antonio FloresMartınez, Maria Fernanda Delfın Ares De Parga and all the other amazing students ofBerlin Mathematical School I did not list. Of course, BMS family consists more thanthe students. I want to particularly thank Annika Preuß-Vermeulen and all the otherBMS staff who helped me whenever I have faced a problem in Berlin. Without theirhelp, my last two years in Berlin would be much harder.

Lastly, I want to thank my professors Prof. Dr. Gulin Ercan, Dr. Ozgur Kisisel andDr. Mehmetcik Pamuk from Middle East Technical University (Turkey) for their endlesssupports during my bachelor studies.

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries 72.1 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Symmetric and Skew-Symmetric Tensors . . . . . . . . . . . . . . . 92.1.3 Young Symmetrizers . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.4 Schur Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Group Actions on Varieties . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.4 The Hilbert-Mumford Criterion . . . . . . . . . . . . . . . . . . . . 252.2.5 Good Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Invariants and Semi-Invariants of Matrices 353.1 Matrix Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.1 A Description of the Nullcone of the Simultaneous Conjugation . 373.1.2 Generators of S(n,m) in Characteristic 0 . . . . . . . . . . . . . . 383.1.3 Generators of S(n,m) in Positive Characteristic . . . . . . . . . . 45

3.2 Matrix Semi-Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.1 Compression Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Tensor Blow-Ups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.3 Generators of R(n,m) . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 Degree Bounds for Generators of S(n,m) and R(n,m) . . . . . . . . . . . 683.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.4.1 The Constructive Regularity Lemma . . . . . . . . . . . . . . . . 723.4.2 The Computation of Non-Commutative Rank . . . . . . . . . . . . 783.4.3 Orbit Closure Problem for the Simultaneous Conjugation . . . . . 853.4.4 Orbit Closure Problem for the Left-Right Action . . . . . . . . . . 90

4 Invariant Theory of Quivers 974.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2 Invariant Theory of Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.2.1 Le Bruyn-Procesi Theorem . . . . . . . . . . . . . . . . . . . . . . 1034.2.2 Donkin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3 Semi-Invariants of Quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.3.1 King’s Stability Condition . . . . . . . . . . . . . . . . . . . . . . . 1104.3.2 Schofield Semi-Invariants . . . . . . . . . . . . . . . . . . . . . . . 114

4.4 Degree Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5 Order Three Tensors 1355.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.2 An Equivariant Map from Tensors to Forms . . . . . . . . . . . . . . . . . 1375.3 Invariants of Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.4 Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.5 The Computation of the Nullcone of Some Spaces . . . . . . . . . . . . . 148

5.5.1 Two Slices Tensors : C2 ⊗ Cn ⊗ Cn . . . . . . . . . . . . . . . . . . 1485.5.2 Cuboids : C3 ⊗ C3 ⊗ C3 . . . . . . . . . . . . . . . . . . . . . . . . 152

Bibliography 157

1 Introduction

“ Whoever thinks algebra is a trick in obtaining unknowns has thoughtit in vain. No attention should be paid to the fact that algebra and ge-ometry are different in appearance. Algebras are geometric facts whichare proved by propositions five and six of Book two of Elements.

”Omar Khayyam (1048-1131),

There are two crucial group actions in linear algebra: The first action is the well-knownconjugation of matrices. More concretely, the group GLn acts on the vector space Matn,nof n× n-matrices via the rule

g ·X = gXg−1.

The other fundamental action is the left-right action given by

(g, h) ·X = gXh−1, (g, h) ∈ GLn ×GLn.

These actions are extensively studied and we can state without hesitation that everythingthat is there to know is known about them.

It is a common theme in invariant theory that even if we know everything about theaction of a group G on a vector space V , we may not understand the action of G onV ⊕m. This is in fact the case for the actions we just mentioned. We encourage theskeptical reader to try to find the normal forms of the action of GLn on pairs of matrices(X,Y ) ∈ Mat2n,n via simultaenously conjugating X and Y . We introduce the followingactions:

1. Simultaneous conjugation. The group GLn acts on tuples of matrices Matmn,n via

g · (X1, . . . , Xm) = (gX1g−1, . . . , gXmg−1).

2. The left-right action. The group GLn×GLn acts on tuples of matrices Matmn,n via

(g, h) · (X1, . . . , Xm) = (gX1h−1, . . . , gXmh−1).

We can see these two actions from another perspective. We can interpret tuples ofmatrices as a collection of linear maps between two vector spaces. Then, the groups

Invariants and Orbit Closure Problems for Quivers and Generalizations to Tensors Mahmut Levent Dogan

of invertible linear endomorphisms of these vector spaces act on the linear maps viaa change of bases. In the case that two vector spaces are equal, the linear maps areendomorphisms and a change of bases corresponds to simultaneous conjugation. If theyare not equal, then it corresponds to the left-right action.

The above interpretation allows further generalization: Assume that V1, . . . , Vn aresome vector spaces and we have a collection of linear maps between them. These linearmaps can possibly be endomorphisms (i.e. a linear map from one of the vector spacesVi to itself) and we might have more than one linear map between two vector spaces.Then, the group of invertible linear endomorphisms act on such linear maps.

We can state the above action more concretely: A quiver Q is a directed graph. LetQ0 denote its vertex set and Q1 denote its edge set. Moreover, for each arrow a ∈ Q1,let ta denote its tail and ha denote its head. To each vertex x ∈ Q0, we assign a vectorspace V (x). Consider the group

G =!

x∈Q0

GL(V (x)).

Then, G acts on "

a∈Q1

Hom(V (ta), V (ha))

via the rule

(g(x) | x ∈ Q0) · (f(a) | a ∈ Q1) = (g(ha)f(a)g(ta)−1 | a ∈ Q1).

Observe that this action generalizes both simultaneous conjugation and the left-rightaction: If Q consists of a single vertex and m loops on it, then the above action isisomorphic to simultaneous conjugation. If Q consists of two vertices and m arrowsfrom the first one to the second one, then it is isomorphic to the left-right action.

There is one last action we will consider. Consider the vector space Matab,c of a-tuplesof b× c-matrices. The group GLb ×GLc acts on Matab,c via the left-right action as it isdescribed above. On the other hand, GLa also acts on Matab,c via the rule

g · (X1, . . . , Xa) = (g11X1 + · · ·+ g1aXa, . . . , ga1X1 + · · ·+ gaaXa)

where gij denotes the (i, j)-th entry of the matrix g. This action is isomorphic to theaction of GLa × GLb × GLc on U ⊗ V ⊗ W where dimU = a, dimV = b, dimW = c.We call an element of U ⊗ V ⊗W an order three tensor. As the advanced reader mightguess, understanding this action is much harder than the case of quivers.

For the actions we described, we are interested in finding the invariants: For a repre-sentation V of the group G, an invariant is a polynomial function

f : V → K

2 Master’s Thesis, TU Berlin, 2019

Mahmut Levent Dogan Invariants and Orbit Closure Problems for Quivers and Generalizations to Tensors

such that f(g · v) = f(v) for all g ∈ G. Here, K denotes the base field. We denotethe set of invariants by K[V ]G and for the actions we consider, it is a finitely generatedK-algebra. Throughout this thesis, we aim to find the invariants of the three actionswe mentioned. If we cannot achieve this goal, we want to describe the zero set of non-constant homogeneous invariants that is called the nullcone.

1.1 Motivation

The actions described above are interesting on their own. However, it turns out thatunderstanding these actions has a wide range of applications. We mention one suchapplication that might interesting to the reader:

Given a polynomial f ∈ K[x1, . . . , xn], in [Val79], Valiant shows the existence of atuple (A1, . . . , An) ∈ Matns,s of matrices such that

f = det(x1A1 + · · ·+ xnAn).

Here, the size of the matrices is poly(n). In particular, the famous problem for deter-mining whether f ≡ 0 (called Polynomial Identity Testing, or PIT for short) can bereduced to determining whether a given subspace 〈A1, . . . , An〉 consists of only singularmatrices. It is shown by Kabanets and Impagliazzo ([KI03]) that the existence of a deter-ministic polynomial time algorithm for PIT has some serious implications in complexitytheory.

There is a similar construction by Cohn in non-commutative case. A non-commutingrational expression φ(x1, . . . , xn) (over the field K) is a formula in non-commuting vari-ables x1, . . . , xn obtained by the arithmetic operations addition, multiplication, inversionand multiplication by scalars in K. For example, xy−yx or (x+xy−1x)−1+(x+y)−1−x−1 are non-commuting rational expression over the non-commuting variables x, y. In[Coh71], Cohn shows that for every non-commuting rational expression φ(x1, . . . , xn),there exists a tuple (A1, . . . , An) ∈ Matns,s for s =poly(n) such that φ(x1, . . . , xn) ≡ 0if and only if (A1, . . . , An) is in the nullcone of the action of SLn × SLn. Recall thatthe nullcone of an action of G on a vector space V is the zero set of all non-constanthomogeneous invariants.

In this thesis, we extensively study the invariant theory of the action of SLn × SLn

on Matmn,n and in particular give a polynomial time algorithm to dedice whether a giventuple (A1, . . . , Am) ∈ Matmn,n is in the nullcone. In particular, this algorithm shows thatwe can decide in polynomial time whether a given non-commuting rational expressionφ(x1, . . . , xm) is zero.

Master’s Thesis, TU Berlin, 2019 3


1.2 Objective

There are two main objectives of this master’s thesis.

Firstly, we wanted to have a treatment of the invariant theory of quiver representationsthat is self-contained and complete. More concretely, we wanted to give all the mainresults on the subject together with their proofs. During the writing of this thesis, werealized that this goal is too ambitious to achieve in a master’s thesis. This is mainlybecause the theory of representations of quivers is deeper than we initially imagined andalso some proofs involve some heavy machinery that is hard to present in a compactway. In order to make this thesis more readable, we had to cut some of the proofs.For example, the proof of Donkin’s theorem in Section 3.1.3 is more of a sketch thana complete proof. The same applies to the degree bound given by Razmyslov. Still,we believe that we partially achieved our original objective as the thesis contains thetheorems we wanted to present and most of their proofs.

Secondly, we wanted to use the ideas of matrix semi-invariants to develop new ideasfor the invariant theory of order three tensors. This is achieved via a special map

ϕ : Ca ⊗ Cn ⊗ Cn → P

(X1, . . . , Xa) )→ det(z1X1 + · · ·+ zaXa)

that we introduce in the last chapter. Here P is the vector space of homogeneouspolynomials of degree d in a variables. The idea to introduce this map comes from apaper by Thrall and Chandler ([TC38]) where the authors describe the GL3×GL3×GL3-orbits of C3 ⊗C3 ⊗C3 via assigning each tensor T a variety X which happens to be thezero set of the polynomial ϕ(T ) that we will define. We could not find a result about thisrelation between order three tensors and matrix semi-invariants in the existing literature,so we consider this last chapter partially original.

1.3 Outline

In Chapter 2, we introduce the reader to our main toolbox under the umbrella of twosections on multilinear algebra and linear algebraic groups. For the first section, wemainly used Landsberg’s book, Tensors : Geometry and Applications [Lan12] and forthe second section we used Humphreys’ book, Linear Algebraic Groups [Hum75]. Inthe end of each section, we added a subsection that introduces the reader to the theoryover the fields of positive characteristic. The construction of Schur functors in posi-tive characteristic mainly follows Akın, Buchsbaum, Weyman [ABW82]. The theory ofgood filtrations and good modules follows Donkin’s book, Rational Representations ofAlgebraic Groups [Don85].

In Chapter 3, we have four main sections. In the first section we study the simultaneousconjugation of tuples of matrices. We start by giving a description of the nullcone of this



action and then turn our attention to finding a generating set of invariants. In [Art69],Artin conjectures that in characteristic 0, the invariant ring of this action is spannedby the traces of words in matrices. We follow the proof of this conjecture by Procesi[Pro76]. Donkin’s generalization ([Don92]) of this theorem to arbitrary characteristic byreplacing traces with characteristic coefficients is given in this section. The second sectionis dedicated to the left-right action on tuples of matrices. Again, we start by giving adescription of the nullcone following a proof by Burgin and Draisma [BD06]. Then, weintroduce the idea of tensor blow-ups and finally we give a description of a generating setof invariants. In the third section, we give degree bounds for a generating set of invariantsand for the invariants that define the nullcone. The main bound for the simultaneousconjugation in characteristic 0 is n2 and it is given by Razmyslov in [Raz74]. In positivecharacteristic, there are polynomial bounds for the simultaneous conjugation and theleft-right action by Derksen and Makam. Lastly, we give the algorithms that solves thecorresponding orbit closure intersection problems for these actions. These algorithms areintroduced by Derksen and Makam in [DM18a]. On the other hand, the algorithm for theleft-right action depends on the algorithm for the computation of the non-commutativerank of matrix subspaces introduced by Ivanyos, Qiao and Subrahmanyam in a series ofpapers.

In Chapter 4, we focus our attention to quiver representations. In the first section,we give the basic definitions. For this section, we mainly used Derksen and Weyman’s[DW17]. The invariant theory of quivers once again splits into two parts: In the secondsection, we study the invariants of quiver representations. Here, we give the generaliza-tion of Procesi’s theorem to arbitrary quivers, called Le Bruyn-Procesi theorem [LP90],which states that the invariant ring is spanned by the traces along oriented cycles un-der the assumption that the characteristic of the field is 0. A similar generalization inpositive characteristic is due to Donkin [Don94]. There, the author proves that the char-acteristic coefficients along oriented cycles span the invariant ring. In the third section,we turn our attention to semi-invariants of quivers. We start by describing semi-stablerepresentations and the nullcone of quiver representations due to King [Kin94]. Then,we introduce Schofield semi-invariants and prove that Schofield semi-invariants span thering of semi-invariants under the assumption that the quiver has no oriented cycles.In the last section, we give degree bounds for generators. This section mainly follows[DM16b] by Derksen and Makam.

The last chapter, Chapter 5, is dedicated to the study of order three tensors. In thissection, start with some preliminaries on the action of SLa×SLb×SLc on Ca⊗Cb⊗Cc.In the second section, we introduce the main tool of the chapter that is an equivariantmap ϕ from Ca ⊗Cn ⊗Cn to Sn((Ca)∗), the forms of degree d in a variables. ϕ is givenby the determinant of the linear matrix z1X1 + · · · + zaXa, where Xi are the slices ofthe tensor in the first direction and zi are formal variables. In the third section, westudy the invariants of forms. The main invariant we consider is the discriminant. Inthe fourth section, we introduce Cayley’s hyperdeterminant and show that its zero setis mapped under ϕ into the singular forms. In the last section, we give a description of



the nullcone of the spaces C2⊗Cn⊗Cn and C3⊗C3⊗C3. Namely, we prove that a twoslices tensor T ∈ C2 ⊗ Cn ⊗ Cn is in the nullcone if and only if the binary form ϕ(T ) isin the nullcone (Theorem 5.5.1) and for a cuboid T ∈ C3 ⊗ C3 ⊗ C3, we show that T isin the nullcone if and only if the ternary cubic ϕ(T ) is in the nullcone and the slices ofT in each direction has a rank 1 section (Theorem 5.5.4).


2 Preliminaries

The aim of this chapter is to introduce the main toolbox of this master’s thesis: Multilin-ear algebra and the representation theory of linear algebraic groups. As it is impossibleto cover everything on these subjects in a master’s thesis, we refer to two introductorybooks on these subjects. The main reference we used for multilinear algebra is JosephLandsberg’s wonderful book [Lan12] and a great introductory book for linear algebraicgroups is James Humphreys’ [Hum75]. In this chapter, we avoid most of the proofs sincethey are either straightforward or can be found easily.

2.1 Multilinear Algebra

Throughout this section, we work over finite dimensional vector spaces over a fixed fieldK. We do not have any assumptions on char(K) unless stated otherwise.

2.1.1 Basic Definitions

Definition 2.1.1. Given vector spaces V1, V2, . . . , Vm,W , a multilinear function is amap

f : V1 × V2 × · · ·× Vm → W

that is linear in each factor, i.e. for all i = 1, . . . ,m we have

f(v1, . . . , vi−1, vi+wi, vi+1, . . . , vm) = f(v1, . . . , vi−1, vi, vi+1, . . . , vm)+f(v1, . . . , vi−1, wi, vi+1, . . . , vm).

We denote the space of all multilinear functions V1× · · ·×Vm → W by V ∗1 ⊗ · · ·⊗V ∗

m⊗Wand call it the tensor product of V ∗

1 , . . . , V∗m,W . In the special case that W = K, we

drop W from the notation and write V ∗1 ⊗ · · · ⊗ V ∗

m. Here, given a vector space V , V ∗

denotes the dual vector space

V ∗ = f : V → K | f is linear.

Given vector spaces V1, . . . , Va,W1, . . . ,Wb, U , the vector space of multilinear func-tions

f : V1 × · · ·× Va ×W1 × · · ·×Wb → U


is naturally isomorphic to the vector space of multilinear functions

g : V1 × · · ·× Va → W ∗1 ⊗ · · ·⊗W ∗

b ⊗ U

viag(v1, . . . , va) = ((w1, . . . , wb) )→ f(v1, . . . , va, w1, . . . , wb)).

Hence no confusion arises from the fact that we use the same notation V ∗1 ⊗ · · ·⊗ V ∗

a ⊗W ∗

1 ⊗ · · ·⊗W ∗b ⊗ U for both of these spaces.

Remark 2.1.2. We use the word naturally to emphasize the fact that this isomorphismdoes not depend on a particular choice of bases for the vector spaces. More formally,this isomorphism is a GL(V1)× · · ·×GL(Va)×GL(W1)× . . . GL(Wb)×GL(U)-moduleisomorphism. Here, V ∗

1 ⊗ · · ·⊗V ∗m is considered as a GL(V1)× · · ·×GL(Vm)-module via

the rule((g1, . . . , gm) · T )(v1, . . . , vm) = T (g−1

1 v1, . . . , g−1m vm).

Definition 2.1.3 (Tensor Rank). Let ϕi ∈ V ∗i be given. We define the multilinear

function

ϕ1 ⊗ · · ·⊗ ϕm : V1 × · · ·× Vm → K

(v1, . . . , vm) )→ ϕ1(v1)ϕ2(v2) . . .ϕm(vm)

so ϕ1 ⊗ · · · ⊗ ϕm ∈ V ∗1 ⊗ · · · ⊗ V ∗

m. An element of V ∗1 ⊗ · · · ⊗ V ∗

m is said to have rank 1if it can be written as above. More generally, the rank of a tensor T ∈ V ∗

1 ⊗ · · · ⊗ V ∗m,

denoted by R(T ), is the minimum number r such that T can be written as the sum ofr rank 1 tensors.

Remark 2.1.4. It is not a priori clear that every tensor T ∈ V1⊗ · · ·⊗Vm can be writtenas a sum of rank 1 tensors. To this end, for each i = 1, . . . ,m fix a basis of Vi and let vijbe the j-th basis vector of Vi. Then

v1j1 ⊗ · · ·⊗ vmjm | j1 ∈ [dimV1], . . . , jm ∈ [dimVm]

is a basis of V1 ⊗ · · · ⊗ Vm. We leave the proof of this fact to the reader. Since eachbasis vector is of rank 1, we deduce that every tensor can be written as a sum of rank 1tensors.

Example 2.1.5. Let V,W be vector spaces. Given linearly independent v1, v2 ∈ V ,w1, w2 ∈ W , the tensor

v1 ⊗ w1 + v2 ⊗ w2

has rank 2, whereasv1 ⊗ w1 + v1 ⊗ w2

has rank 1 since it equals

v1 ⊗ w1 + v1 ⊗ w2 = v1 ⊗ (w1 + w2).

This equality follows by multilinearity.



Proposition 2.1.6. The rank of a tensor T ∈ V1 ⊗ · · · ⊗ Vm is invariant under theaction of G = GL(V1)× · · ·×GL(Vm), i.e. if g = (g1, . . . , gm) ∈ GL(V1)× · · ·×GL(Vm),then R(T ) = R(g · T ) for all T ∈ V1 ⊗ · · ·⊗ Vm.

Proof. Firstly, if T = v1⊗ · · ·⊗ vr is a rank 1 tensor, then for all g = (g1, . . . , gm) ∈ G =GL(V1)× · · ·×GL(Vm) we have

g · T = (g1v1)⊗ · · ·⊗ (gmvm)

so R(g · T ) = 1. Conversely, if R(g · T ) = 1, then R(T ) = R(g−1 · (g · T )) = 1 by theabove reasoning. Thus the theorem holds for rank 1 tensors.

The action ofG on V is linear : for all g ∈ G and v, w ∈ V we have g·(v+w) = g·v+g·w.Hence if T = v1+ · · ·+vr is the sum of r rank 1 tensors, then g ·T = g ·v1+ · · ·+g ·vr andeach g · vi has rank 1. Thus R(g · T ) ≤ R(T ). Conversely, if R(g · T ) = r then R(T ) =R(g−1 · (g · T )) ≤ R(g · T ) by the same reasoning. Thus we obtain R(T ) = R(g · T ).

Definition 2.1.7 (Modes of a Tensor). Let T ∈ V1 ⊗ · · ·⊗ Vm be given. Then T can beidentified with a linear map

T : V ∗i → V1 ⊗ · · ·⊗ Vi−1 ⊗ Vi+1 ⊗ · · ·⊗ Vm

ϕ )→#(ϕ1, . . . ,ϕi−1,ϕi+1, . . . ,ϕm) )→ T (ϕ1, . . . ,ϕi−1,ϕ,ϕi+1,ϕm)

$

which is called the i-th mode of T . The columns of the matrix of T are called the slicesin the i-th direction.

Example 2.1.8. Let V,W,U of dimension α,β, γ be given and let V have a basis v1, . . . , vαand similarly W and U . Then every tensor T ∈ V ⊗W ⊗ U can be written as

T i,j,kvi ⊗ wj ⊗ uk

This allows us to identify T with a three dimensional α × β × γ-array where (i, j, k)-thentry is T i,j,k. With this identification, the first mode of T is is a βγ − α-matrix givenby

T =

%

&&&&'

T 1,1,1 T 2,1,1 . . . Tα,1,1

T 1,1,2 . . ....

.... . .

...T 1,β,γ . . . . . . Tα,β,γ

(

))))*

2.1.2 Symmetric and Skew-Symmetric Tensors

Throughout, we assume that char(K) = 0 and use the notation V ⊗m = V ⊗ · · ·⊗ V forthe m-fold tensor product of V .



Definition 2.1.9. A tensor T ∈ V ⊗m is called a symmetric tensor if

T (ϕσ(1), . . . ,ϕσ(m)) = T (ϕ1, . . . ,ϕm)

for all ϕi ∈ V ∗ and σ ∈ Sm. We denote the set of symmetric tensors with Sm(V ).Similarly, we call T skew-symmetric if

T (ϕσ(1), . . . ,ϕσ(m)) = sgn(σ)T (ϕ1, . . . ,ϕm)

for all ϕi ∈ V ∗ and σ ∈ Sm. Here, sgn(σ) denotes the sign (or parity) of σ whichequals (−1)n if σ can be written as a product of n transpositions. We denote the set ofskew-symmetric tensors with Λm(V ).

Proposition 2.1.10. Sm(V ) and Λm(V ) are GL(V )-submodules of V ⊗m and if m ≥ 2then they have trivial intersection.

Proof. Assume that g ∈ GL(V ) and T ∈ Sm(V ). Then

(g · T )(ϕσ(1), . . . ,ϕσ(m)) = T (g−1ϕσ(1), . . . , g−1ϕσ(m))

= T (g−1ϕ1, . . . , g−1ϕm)

= (g · T )(ϕ1, . . . ,ϕm)

Thus Sm(V ) is GL(V )-stable. Similarly one can show that Λm(V ) is GL(V )-stable. Itis easy to show that both Sm(V ) and Λm(V ) are subspaces.

Say m ≥ 2. Sm(V ) and Λm(V ) have trivial intersection as if T ∈ Sm(V )∩Λm(V ) then

T = (12) · T = sgn((12)) · T = −T

so T = 0.

Example 2.1.11. Let m = 2. Then we have

S2(V ) = 〈v ⊗ w + w ⊗ v | v, w ∈ V 〉= 〈v ⊗ v | v ∈ V 〉

Λ2(V ) = 〈v ⊗ w − w ⊗ u | v, w ∈ V 〉

Moreover, we have

V ⊗ V = S2(V )⊕ Λ2(V )

since for any rank 1 tensor we have v⊗w = 12(v⊗w+w⊗v)+ 1

2(v⊗w−w⊗v). Observethat the first summand is symmetric and the second summand is skew-symmetric. Weencourage the reader to compare this to the well-known linear algebra fact : Everysquare matrix can be written as the sum of a symmetric matrix and a skew-symmetricmatrix.



For m > 2, we define

πS : V ⊗m → V ⊗m

v1 ⊗ · · ·⊗ vm )→ v1v2 . . . vm =1

m!

+

σ∈Sm

vσ(1) ⊗ · · ·⊗ vσ(m),

πΛ : V ⊗m → V ⊗m

v1 ⊗ · · ·⊗ vm )→ v1 ∧ v2 ∧ · · · ∧ vm =1

m!

+

σ∈Sm

sgn(σ)vσ(1) ⊗ · · ·⊗ vσ(m)

πS is called the symmetrization map and πΛ is called the skew-symmetrization map.Then we have Sm(V ) = πS(V

⊗m) and Λm(V ) = πΛ(V⊗m).

The following lemma is well-known (see [Lan12] Section 2.6.4 for example).

Lemma 2.1.12. Let K[V ]d denote the space of polynomial maps V → K of degree d.The map

ϕ : Sd(V ∗) → K[V ]d

T )→ (x )→ T (x, x, . . . , x))

is a GL(V )-module isomorphism.

The inverse of ϕ is called polarization. Given a polynomial f ∈ K[V ]d, the polarizationgives a symmetric tensor T ∈ Sd(V ∗) via the polarization identity

T (v1, . . . , vd) =1

k!

+

I⊆[d],I ∕=∅

(−1)d−|I|f(+

i∈Ivi).

Example 2.1.13. Consider the polynomial f = x2y ∈ K[x, y]. The polarization identitygives

T (v1, v2, v3) =1

2(f(v1+v2+v3)−f(v1+v2)−f(v1+v3)−f(v2+v3)+f(v1)+f(v2)+f(v3))

so T = 13(x1x2y1 + x1x3y2 + x2x3y1).

2.1.3 Young Symmetrizers

We have already shown that V ⊗2 splits as a direct sum

V ⊗2 = S2(V )⊕ Λ2(V ).

For m > 2, there is a way to write V ⊗m as a direct sum of irreducible GL(V )-modulesusing Young symmetrizers. First, we define what a partition is:



Definition 2.1.14. Let m ∈ Z>0 be given. A tuple λ = (λ1, . . . ,λk) ∈ Z≥0 is called apartition of m if

,i λi = m and λ1 ≥ λ2 ≥ · · · ≥ λk. In this case we write λ ⊢ m.

It is customary to denote partitions via Young diagrams. A Young diagram associatedto a partition λ = (λ1, . . . ,λk) is a collection of boxes such that there are λi boxes inthe i-th row, as in the following example

which is the Young diagram corresponding to the partition (4, 3, 1) ⊢ 8.

Definition 2.1.15. A Young tableau corresponding to a partition λ ⊢ m is a filling ofthe Young diagram of λ with elements of [m]. If each number is used once, then we callit a Young tableau without repetition.

Example 2.1.16. Let m = 3. Then m has 3 partitions and all possible Young tableauxwithout repetition are given as follows :

1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1

1 23

1 32

2 13

2 31

3 12

3 21

123

132

213

231

312

321

Definition 2.1.17. Let λ = (λ1, . . . ,λk) ⊢ m and assume that Tλ is a Young tableauof λ without repetition. Let Sti and Stj denote the permutation groups of the i-th rowand j-th column respectively. For each row and column of Tλ we define

ρti =+

g∈Sti

eg ∈ K[Sm],

ρtj =+

g∈Stj

sgn(g)eg ∈ K[Sm].

We define the Young symmetrizer of Tλ to be

ρTλ= ρt1ρt2 . . . ρtkρt1 . . . ρtλ1 ∈ K[Sm]

Note that since ρTλ∈ K[Sm], it can be multiplied with any element of an Sm-module.

In particular we have the following definition:

Definition 2.1.18. Let V be a vector space. Given λ ⊢ m and a Young tableau Tλ ofλ without repetition, define

LTλ= ρTλ

(V ⊗m)



The following are well-known:

Proposition 2.1.19. Let Tλ, T′λ be two Young tableaux of the partition λ without repe-

tition. Then LTλand LT ′

λare isomorphic GL(V )-modules. Hence, we let Lλ(V ) denote

any of the LTλ(V ). Moreover, if Lλ(V ) ∕= 0, then it is an irreducible GL(V )-module.

Theorem 2.1.20. V ⊗m has a decomposition into irreducible GL(V )-modules as

V ⊗m ="

λ⊢m(Lλ(V ))⊕fλ

where fλ are some natural numbers.

Example 2.1.21. Assume that V is an n-dimensional vector space. In the special caseλ = (n) ⊢ n, we have

Lλ(V ) = Sn(V )

and in the case λ = (1, 1, . . . , 1) ⊢ n we have

Lλ(V ) = Λn(V ).

In fact, a more general theorem called the Schur-Weyl duality holds:

Theorem 2.1.22 (Schur-Weyl Duality). V ⊗m has a decomposition into GL(V ) × Sm

-irreducible modules given by

V ⊗m ="

λ⊢m,l(λ)≤dimV

Lλ(V )⊗ Specht(λ)

where l(λ) is the number of rows in the Young diagram of λ and Specht(λ) is an irre-ducible Sm-module called the Specht module corresponding to the partition λ.

Alternatively, Schur-Weyl duality states that in the algebra Hom(V ⊗m, V ⊗m), thegroups GL(V ) and Sm are pairwise centralizers of each other.

Remark 2.1.23. The above theorems give a decomposition of V ⊗m as a GL(V )-module.When V = U ⊗W , one may want to decompose modules with respect to the action ofGL(U)×GL(W ). As only modules that appear in the decomposition of V ⊗m are of theform Lλ(V ), we only need to decompose Lλ(U⊗W ) as a GL(U)×GL(W )-module. Thistask is not easy in general, but in the special case that λ = (n) or λ = (1n), followingformulas are known:

Theorem 2.1.24 (Cauchy Formulas). Let V,W be two vector spaces. Then Sn(V ⊗W )and Λn(V ⊗W ) has the decomposition

Sn(V ⊗W ) ="

λ⊢nLλ(V )⊗ Lλ(W )

Λn(V ⊗W ) ="

λ⊢nLλ(V )⊗ Lλ′(W )



as a GL(V )×GL(W )-module.

Here, λ′ denotes the conjugate partition of λ, that has the Young diagram obtained byreflecting the Young diagram of λ.

2.1.4 Schur Functors

Observe that if K is a field of positive characteristic, we may not be able to define thesymmetrization and the skew-symmetrization maps as their definition require divisionby an integer. In this section we try to outline the theory of Schur functors in positivecharacteristic. We follow the paper by Akın, Buchsbaum and Weyman [ABW82].

Throughout, K denotes an algebraically closed field of arbitrary characteristic.

Definition 2.1.25. Let V be a vector space. Define the space

T (V ) ="

d≥0

V ⊗d

where V ⊗0 is defined to be K. T (V ) admits a natural graded K-algebra structure wherethe multiplication is given by

(v1 ⊗ · · ·⊗ vs) · (w1 ⊗ · · ·⊗ wt) = v1 ⊗ · · ·⊗ vs ⊗ w1 ⊗ · · ·⊗ wt

on rank 1 tensors.

We define two ideals of T (V ):

I = T (V ) · v ⊗ v | v ∈ V

andJ = T (V ) · v ⊗ w − w ⊗ v | v, w ∈ V

We define the exterior algebra Λ(V ) and the symmetric algebra S(V ) to be

Λ(V ) = T (V )/I, S(V ) = T (V )/J

we denote the image of v1⊗ · · ·⊗ vk under these quotients by v1∧ · · ·∧ vk and v1v2 . . . vkrespectively.

Moreover, as I and J have a set of homogeneous generators, they also inherit thegrading of T (V ):

Λ(V ) ="

d≥0

Λd(V ), S(V ) ="

d≥0

Sd(V ).

Given a vector of natural numbers λ = (λ1, . . . ,λk) ∈ Zk≥0, we define

Sλ(V ) = Sλ1(V )⊗ Sλ2(V )⊗ · · ·⊗ Sλk(V )



and similarly

Λλ(V ) = Λλ1(V )⊗ Λλ2(V )⊗ · · ·⊗ Λλk(V )

In order to define the Schur modules, for each partition λ ⊢ n, we want to define a mapdλ(V ) : Λλ(V ) → Sλ′

(V ) where λ′ denotes the conjugate partition of λ.

Definition 2.1.26 (Comultiplication). Given V , consider the diagonal embedding

ϕ : V → V ⊕ V

v )→ (v, v).

The embedding ϕ induces a K-algebra map

∆ : Λ(V ) → Λ(V )⊗ Λ(V )

v1 ∧ · · · ∧ vk )→+

0≤s≤k

+

σ

sgn(σ)vσ(1) ∧ · · · ∧ vσ(s) ⊗ vσ(s+1) ∧ · · · ∧ vσ(k)

where the second sum runs over all permutations σ ∈ Sk satisfying σ(1) < σ(2) < · · · <σ(s) and σ(s+ 1) < · · · < σ(k). We call ∆ the comultiplication map of Λ(V ).

In particular we have

∆ : Λ1(V ) = V → Λ(V )⊗ Λ(V )

v )→ v ⊗ 1 + 1⊗ v

and

∆ : Λk(V ) →k"

s=0

Λs(V )⊗ Λk−s(V ).

Definition 2.1.27. Given a partition λ = (λ1, . . . ,λk) ⊢ m, let µ = (µ1, . . . , µk) denotethe partition obtained by deleting the first column of the Young diagram of λ. Moreover,let λ′ = (λ′

1, . . . ,λ′l(λ)) be the conjugate partition of λ. Define the map δλ to be the

composition of maps

Λλ1(V )⊗ · · ·⊗ Λλk(V ) V ⊗ Λλ1−1(V )⊗ V ⊗ Λλ2−1(V )⊗ · · ·⊗ V ⊗ Λλk−1(V )

V ⊗k ⊗ Λµ(V ) Sk(V )⊗ Λµ(V ).∼= π⊗1

∆⊗···⊗∆

Here, each Λλi(V ) is mapped to

λi"

s=0

Λs(V )⊗ Λk−s(V )

which surjects onto Λ1(V )⊗Λλ1−1(V ) = V ⊗Λλ1−1(V ). With a slight abuse of notation,we denote the composition of ∆ with this surjection again with ∆.



Now we can inductively define dλ(V ). For λ = (0), we define

dλ(V ) : Λλ(V ) = K → K = Sλ(V )

to be the identity map.

Assume that λ ⊢ n and for all m < n and µ ⊢ m, the map dµ(V ) is defined. We definedλ(V ) to be the composition

Λλ(V ) Sk(V )⊗ Λµ(V ) Sk(V )⊗ Sµ′(V ) = Sλ′

(V ).1⊗dµ(V )δλ

We define the Schur functor Lλ′(V ) as the image of dλ(V ):

Definition 2.1.28. The image of dλ(V ) is called the Schur functor corresponding tothe partition λ′ and denoted by Lλ′(V ).

Warning: In [ABW82], the authors define Lλ(V ) to be the image of dλ(V ). In orderto make the definitions consistent with the previous chapter, we define Lλ(V ) to be theimage of dλ′(V ). Thus, to us, L(n)(V ) is isomorphic to Sn(V ) whereas for the authorsof [ABW82], it is isomorphic to Λn(V ).

Example 2.1.29. In characteristic 0, the construction of Lλ(V ) using Young symmetrizersand the construction above agrees. As an example we have

d(2)(V ) : Λ2(V ) → S(1,1)(V ) = V ⊗ V

v1 ∧ v2 )→ v1 ⊗ v2 − v2 ⊗ v1

so the image of d(2)(V ) is Λ2(V ) = L(1,1)(V ).

Theorem 2.1.30 (The First Cauchy Formula). Let k ∈ Z≥1 and let λ1, . . . ,λt be acomplete set of partitions of k.

There is a filtration

0 = W0 ⊆ W1 ⊆ · · · ⊆ Wt = Sk(V ⊗W )

such that

Wi+1/Wi∼= Lλi+1

(V )⊗ Lλi+1(W )


The second Cauchy formula requires more work.

Definition 2.1.31. We define the divided power algebra D(V ) to be the graded dualof S(V ∗), i.e.

D(V ) = S(V ∗)∗.



Remark 2.1.32. In characteristic 0, there is a natural identification S(V ∗)∗ = S(V ) givenby

K[x1, . . . , xn]∗ ∼= K[∂1, . . . , ∂n]

where ∂i denotes the formal partial derivative in the i-th direction.

In positive characteristic this no longer holds as if char(K) = p, then ∂pi ≡ 0. Instead,

given α = (α1, . . . ,αn) ∈ Zn we define the operator

D(α)(xβ11 . . . xβn

n ) = (

n!

i=1

-βiαi

.)xβ1−α1

1 . . . xβn−αnn

Observe thatα1! . . .αn!D

(α) = ∂α = ∂α11 . . . ∂αn

n

Then D(α) | α ∈ Zn form a basis for D(V ) that is dual to the basis xα11 , . . . , xα

n

n | α ∈Zn of S(V ∗). In particular, for α = (0, . . . , 0, 1, 0, . . . , 0) where 1 is in the j-th position,

we have D(α) = vj where vj is the j-th basis vector of V . Setting v(p)j = Dp·α, we have

p!v(p)j = vpj ∈ Sp(V ).

Definition 2.1.33. We define the comultiplication of the divided power algebra to be

∆DF : D(V ) → D(V )⊗D(V )

v(α1)1 . . . v(αn)

n )→+

0≤βi≤αi

v(β1)1 . . . v(βn)

n ⊗ v(α1−β1)1 . . . v(αn−βn)

n .

Similar to the previous case, we can define a map δ′λ to be the composition

Dλ1(V )⊗ · · ·⊗Dλk(V ) V ⊗Dλ1−1(V )⊗ V ⊗Dλ2−1(V )⊗ · · ·⊗ V ⊗Dλk−1(V )

V ⊗k ⊗Dµ(V ) Λk(V )⊗Dµ(V )∼= π⊗1

∆⊗···⊗∆

and constructd′λ : Dλ(V ) → Λλ′

(V ).

Definition 2.1.34. The image of d′λ(V ) is called the coSchur functor corresponding tothe partition λ′ and denoted by Kλ′(V ).

Remark 2.1.35. In characteristic 0, we have

Kλ(V ) ∼= Lλ′(V ).

Theorem 2.1.36 (The Second Cauchy Formula). Let k ∈ Z≥1 and let λ1, . . . ,λt be acomplete set of partitions of k.

There is a filtration

0 = U0 ⊆ U1 ⊆ · · · ⊆ Ut = Λk(V ⊗W )



such that

Ui+1/Ui∼= Lλi+1

(V )⊗Kλi+1(W )


2.2 Linear Algebraic Groups

In this section, we try to give a basic introduction to the theory of linear algebraicgroups. For more information, we refer to [Hum75].

We fix an algebraically closed field K (of arbitrary characteristic) and assume thatevery variety is over K.

2.2.1 Basic Definitions

Definition 2.2.1. Let G be a variety (possibly not irreducible) endowed with a groupstructure. If the maps

µ : G×G → G

(g, h) )→ gh

ı : G → G

g )→ g−1

are morphism of varieties, then G is called a linear algebraic group.

A morphism of linear algebraic groups is a morphism of varieties ϕ : G → H that isalso a group homomorphism. Similarly, we define isomorphisms and automorphisms oflinear algebraic groups.

Example 2.2.2. 1. The underlying field K is a linear algebraic group with group lawµ(g, h) = g + h. Similarly, K× = K − 0 is a linear algebraic group with grouplaw µ(g, h) = gh.

2. Let GLn denote the set of invertible n× n-matrices. This is a group under matrixmultiplication. Let V = Matn,n denote the vector space of n × n-matrices. ThenGLn is the principal open subset of V corresponding to the polynomial det ∈ K[V ].Hence GLn has an affine variety structure. The matrix multiplication and thematrix inversion are morphisms of varieties, though the latter one is not a prioriclear.

3. The group SLn consists of n×n-matrices of determinant 1. The group Tn is the setof all invertible diagonal n × n-matrices and Un is the set of all upper triangularmatrices with diagonal entries equal to 1. All these groups are linear algebraicgroups and they are closed subvarieties of GLn.



4. All finite groups admit a linear algebraic group structure by being endowed withthe discrete topology.

2.2.2 Group Actions on Varieties

Let G be a linear algebraic group and X be an affine variety. We say that G actsmorphically on X if there exists a morphism of varieties

ϕ : G×X → X

satisfying

ϕ(gh, x) = ϕ(g,ϕ(h, x)) for all g, h ∈ G, x ∈ X

ϕ(1, x) = x for all x ∈ X.

From now on, we use the simpler notation g · x = ϕ(g, x). Given an element x ∈ X, theorbit of x is defined to be

G · x = g · x | g ∈ G.

As a special case of a group acting morphically on a variety, we have rational represen-tations:

Definition 2.2.3. Let V be a vector space and G be a linear algebraic group. We callV a rational G-module if G acts morphically on V with the additional assumptions that

g · (v + w) = g · v + g · w for all g ∈ G, v, w ∈ V

and

g · (λv) = λg · v for all g ∈ G, v ∈ V,λ ∈ K.

In other words, V is called a rational G-module if V is a G-module such that the actionof G on V is a morphism of varieties. If V is a rational G-module then there is aninduced morphism of linear algebraic groups (called a rational representation)

ρ : G → GL(V )

such that g · v = ρ(g)v.

A subspace U ⊆ V is called a G-submodule if g · u ∈ U for all u ∈ U .

V is called an irreducible G-module if only G-submodules are 0 and V .

Recall that for a finite group G of order not divisible by char(K), Maschke’s theorem(see, for example, [Isa06], Theorem 1.9) implies that every G-module V can be writtenas a direct sum of irreducible submodules. For infinite groups, however, there mightbe G-modules that does not split into a direct sum of irreducible submodules. Thisobservation leads to the following definition:



Definition 2.2.4. A linear algebraic group G is said to be linearly reductive if everyrational G-module can be written as the direct sum of irreducible submodules.

Example 2.2.5. In characteristic 0, the groups GLn, SLn, Tn and finite groups are linearlyreductive. Un, on the other hand, is not linearly reductive.

If char(K) ∕= 0, GLn and SLn are no longer linearly reductive and a finite group islinearly reductive if and only if its order is not divisible by char(K).

Even though GLn is not linearly reductive in positive characteristic, it imitates someof the nice properties of linearly reductive groups.

Definition 2.2.6. A unipotent group G is a linear algebraic group that is isomorphicto a closed subgroup of Un.

Definition 2.2.7. A connected linear algebraic group G is called reductive if everyclosed, connected, normal, unipotent subgroup of G is trivial.

Example 2.2.8. • The additive group K is not reductive as K = U1.

• GLn and SLn are reductive.

Remark 2.2.9. When char(K) = 0, a group G is reductive if and only if it is linearlyreductive. The distinction appears in positive characteristic as for example GLn isreductive but not linearly reductive.

Proposition 2.2.10. Every connected linearly reductive group is reductive.

Some properties of reductive groups is given in the next section.

2.2.3 Invariant Theory

In the rest of the chapter, we assume that all of our groups are linear algebraic groupsover an algebraically closed field K.

Definition 2.2.11. Let V be a rational representation of the group G. Then there isan induced action of G on K[V ] via

(g · f)(v) = f(g−1 · v).

A polynomial f ∈ K[V ] is called an invariant if

g · f = f for all g ∈ G.

We denote the set of invariants by K[V ]G.

Example 2.2.12. Sn acts on Kn via

σ · (v1, . . . , vn) = (vσ−1(1), . . . , vσ−1(n)).



The following polynomials are invariant under the induced action of G on K[x1, . . . , xn]and they are called the elementary symmetric polynomials:

e1 =x1 + x2 + · · ·+ xn

e2 =x1x2 + x1x3 + · · ·+ xn−1xn...

ed =+

1≤i1≤i2≤···≤id≤n

xi1xi2 . . . xid

...

en =x1x2 . . . xn.

The Fundamental Theorem of Symmetric Polynomials state that every invariant of thisaction can be written as a polynomial in ei: For each F ∈ K[x1, . . . , xn]

Sn , there existsG ∈ K[y1, . . . , yn] such that

F (x1, . . . , xn) = G(e1, . . . , en).

Alternatively, we haveK[x1, . . . , xn]

Sn = K[e1, . . . , en].

The invariants have the following property:

Proposition 2.2.13. Invariants are constant on the orbits, i.e. if f ∈ K[V ]G andw ∈ G · v = g · v | g ∈ G, then f(v) = f(w).

Proof. Let w = g · v. Then we have

f(w) = f(g · v) = (g−1 · f)(v) = f(v).

Under some assumptions, the converse of the above proposition also holds. We startwith the introduction of the problems in which we are interested.

Orbit Intersection Problem: Given an action of the group G on a set Ω, the orbitintersection problem is the decision problem with input x, y ∈ Ω which outputs whetherG · x ∩G · y = ∅. Note that

G · x ∩G · y ∕= ∅ ⇐⇒ x ∈ G · y ⇐⇒ y ∈ G · x ⇐⇒ G · x = G · y.

As an instance of orbit intersection problem we have the graph isomorphism problemwhere the group Sn acts on Matn,n.

When G is a linear algebraic group and V a rational representation, we may use theinvariants to separate orbits by evaluating the invariants on two points: If they give



different values, then the points are not in the same orbit. The converse does not holdsince given v ∈ V a set of the form

w ∈ V | f(w) = f(v)

is necessarily Zariski closed whereas orbits may or may not be closed. This observationleads to the following problem:

Orbit Closure Intersection Problem: Given a linear algebraic group G and arational representation V , the orbit closure intersection problem is the decision problemwith input v, w ∈ V that outputs whetherG · x∩G · y = ∅. Under additional hypotheses,the orbit closure intersection problem is closely related to the invariant theory as thefollowing theorem suggests:

Proposition 2.2.14 ([MFK94]). Let G be a reductive group and V be a rational repre-sentation. Given two vectors v, w ∈ V , we have

G · v ∩G · w = ∅ ⇐⇒ ∃f ∈ K[V ]G, f(v) ∕= f(w).

The theorem above is not a priori efficient as K[V ]G is an infinite set. However, ifK[V ]G is finitely generated as a K-algebra then evaluating points on a set of generatorsis enough:

Theorem 2.2.15. Let G be a reductive group and V be a rational representation. More-over assume that K[V ]G = K[f1, . . . , fr] for some f1, . . . , fr ∈ K[V ]. Given two vectorsv, w ∈ V , we have

G · v ∩G · w = ∅ ⇐⇒ ∃i = 1, . . . , r, fi(v) ∕= fi(w).

In Example 2.2.12, we are able to find polynomials f1, . . . , fr such that K[V ]G =K[f1, . . . , fr]. This is not always possible but for most of the actions we consider, thisis the case as the following theorem shows:

Theorem 2.2.16. Let V be a rational representation of a reductive group G. ThenK[V ]G is finitely generated, i.e. there exist f1, . . . , fr such that

K[V ]G = K[f1, . . . , fr].

Remark 2.2.17. Even though this theorem guarantees the existence of a finite set ofgenerators for K[V ]G, this set is usually very large to use in practice. Even in smalldimensions, we do not know a set of generators for many actions.

Definition 2.2.18. Let G be a reductive group and V be a rational representation.Suppose that f1, . . . , fr is a finite set of generators for the invariant ring, i.e. K[V ]G =K[f1, . . . , fr]. Consider the map

ψ : V → Kr

v )→ (f1(v), . . . , fr(v)).



Then ψ : V → Im(ψ) is called the categorical quotient of the action of G on V . Wedenote Im(ψ) via V//G. Observe that V//G satisfies K[V//G] = K[f1, . . . , fr].

Definition 2.2.19. A group character σ is a morphism of linear algebraic groups

σ : G → K×.

We call a homogeneous polynomial f ∈ K[V ] a semi-invariant if there exists a groupcharacter σ : G → K× with the property that

g · f = σ(g)f, for all g ∈ G.

In this case we call f a semi-invariant of weight σ. We use the notations

SI(V,σ) = f ∈ K[V ] | f is a homogeneous semi-invariant of weight σ,

K[V ]G,σ ="

n∈Z≥0

SI(V,σn).

Remark 2.2.20. There are two different usages of the term weight in invariant theory.Some authors call a polynomial f a semi-invariant of weight σ if

f(g · v) = σ(g)f(v) for all g ∈ G, v ∈ V.

With respect to Definition 2.2.19, we call such an f a semi-invariant of weight σ−1, whereσ−1 is the group character

σ−1 : G → K×

g )→ (σ(g))−1.

There is a way to see semi-invariants of a given weight as the invariants of anothermodule:

Definition 2.2.21. Let σ : G → K× be a group character. We define a one-dimensionalG-module Kσ where Kσ = K as a vector space and the action of G is given by

g · λ = σ(g)λ.

Proposition 2.2.22. Let G be a reductive group and V be a rational representation.Given a group character σ, define

ϕσ : V → V ⊗Kσ

v )→ v ⊗ 1.

Thenϕ( : K[V ⊗Kσ]G → K[V ]G,σ

is an isomorphism of K-algebras.



Proof. Assume that f ∈ K[V ⊗ Kσ]G is a homogeneous invariant of degree d. Giveng ∈ G we have

(g · ϕ(f)(v) = (ϕ(f)(g−1v)

= f(g−1v ⊗ 1)

= f/σ(g)(g−1v ⊗ g−11)

0

= σ(g)df(g−1 · (v ⊗ 1))

= σ(g)d(g · f)(v ⊗ 1)

= σ(g)df(v ⊗ 1)

= σ(g)d(ϕ(f)(v).

Thus, ϕ( maps the homogeneous invariants of degree d to the semi-invariants of weightσd. Moreover, ϕ( is surjective as given a semi-invariant F ∈ K[V ] of weight σ, thepolynomial

f : V ⊗Kσ → K

v ⊗ λ )→ λF (v)

is an invariant and satisfies ϕ(f = F .

Classical Invariant Theory: Given a rational module V of a reductive group G, atheorem that gives a set of generators for K[V ]G is called the first fundamental theoremof invariant theory for this action. We have the following classical results for GL(V ) andSL(V ):

Theorem 2.2.23 (The First Fundamental Theorem of Invariant Theory for GL(V )).The ring of invariants of the action of GL(V ) on V p⊕ (V ∗)q is spanned by the elementsof the form

〈i | j〉 : V p ⊕ (V ∗)q → K

(v1, . . . , vp,ϕ1, . . . ,ϕq) )→ ϕj(vi).

We can also give a geometric interpretation to the First Fundamental Theorem of In-variant Theory forGL(V ): TheGL(V ) module V p is naturally isomorphic to Hom(Kp, V )and (V ∗)q is naturally isomorphic to Hom(V,Kq). Consider the map

ψ : Hom(Kp, V )×Hom(V,Kq) → Matq,p

(f, g) )→ g f.

The coordinate ring of Matq,p can be identified with K[xi,j | i = 1, . . . , q, j = 1, . . . , p].Then the induced ring map ψ( maps xi,j to ϕi(vj). In fact, the image of the map ψ isΣdimV ⊆ Matq,p where Σk denotes the matrices of rank at most k. Then

ψ : Hom(Kp, V )×Hom(V,Kq) → Σk

is the categorical quotient map.



Theorem 2.2.24 (The First Fundamental Theorem of Invariant Theory for SL(V )).The ring of invariants of the action of SL(V ) on V p⊕ (V ∗)q is spanned by the elementsof the form

〈i | j〉 : V p ⊕ (V ∗)q → K

(v1, . . . , vp,ϕ1, . . . ,ϕq) )→ ϕj(vi)

and (in the case of p ≥ dimV or q ≥ dimV ) the elements of the form

deti1,...,idimV: V p ⊕ (V ∗)q → K

(v1, . . . , vp,ϕ1, . . . ,ϕq) )→ det(1vi1 vi2 . . . vidimV

2)

and

detj1,...,jdimV : V p ⊕ (V ∗)q → K

(v1, . . . , vp,ϕ1, . . . ,ϕq) )→ det(1ϕj1 ϕj2 . . . ϕjdimV

2).

Here, vi and ϕj are considered as column vectors.

Remark 2.2.25. Observe that the First Fundamental Theorem of Invariant Theory forGL(V ) implies that in the case of q = 0, there are no GL(V )-invariants. Similarly, forSL(V ), in the case of q = 0 and p < dimV there are no invariants either.

2.2.4 The Hilbert-Mumford Criterion

Let G be a linear algebraic group and V be a rational representation. Set

∆ = g ∈ G | g · v = v for all v ∈ V

to be the kernel of the representation. A point x ∈ V is called

• unstable if 0 ∈ G · x,

• semi-stable if x is not unstable, i.e. 0 ∕∈ G · x,

• stable if x ∕= 0, the orbit G · x is closed and the group quotient

StabG(x)/∆ = g +∆ | g · x = x

is finite (or equivalently, dimG · x = dimG/∆).

We denote the sets of unstable, semi-stable and stable points by NG(V ), V ss, V s, re-spectively. We call the action of G on V semi-stable (respectively stable) if V ss ∕= ∅(respectively V s ∕= ∅). Otherwise, we call it unstable. We call the set of unstable points,NG(V ), the nullcone of the action. We have the following proposition:

Proposition 2.2.26. Assume that G is reductive. The nullcone NG(V ) is the zero-setof all non-constant homogeneous invariants.



Proof. Let x ∈ NG(V ). Then by definition we have 0 ∈ G · x, which readily implies that

f(x) = f(0) = 0

for all non-constant homogeneous invariants f ∈ K[V ]. The converse follows by Propo-sition 2.2.14.

Example 2.2.27. 1. Consider the action of GLn on Kn. Given any non-zero vectorsv, w ∈ GLn, there exists g ∈ GLn such that gv = w. Hence the orbits of thisaction are 0 and Kn − 0. As Kn − 0 = Kn, we deduce that the action ofGLn on Kn is unstable.

2. SLn acts on (Kn)n via

g · (v1, . . . , vn) = (gv1, . . . , gvn).

In coordinates, this action corresponds to SLn acting on Matn,n via left multipli-cation. By the First Fundamental Theorem of Invariant Theory for SLn, we knowthat the invariant ring is spanned by the determinant. We deduce that the actionof SLn on Matn,n is semi-stable since 0 ∕∈ GLn ·X if detX ∕= 0. It is even stablesince

SLn · In = SLn ⊆ Matn,n

is closed and has dimension n2 − 1 = dimSLn.

3. Assume that n ≥ 2 and consider the action of GLn on Matn,n via conjugation.This action is semi-stable as det is a non-constant invariant. However, the actionis not stable: The conjugation does not change the determinant and the trace ofa matrix so the orbits have codimension at least 2 (so dimension at most n2 − 2).However, ∆ = λIn | λ ∈ K× has dimension 1 so GLn/∆ has dimension n2 − 1.For n = 1, the action of GL1 = K× on K is trivial so in this case the action isstable.

The Hilbert-Mumford Criterion is a tool to compute the nullcone NG(V ) withoutrelying on the invariant ring. First, we need a setup:

Definition 2.2.28. A one-parameter subgroup of a linear algebraic group G is a mor-phism of linear algebraic groups

λ : K× → G.

Given a rational G-module V and a vector v ∈ V , a one-parameter subgroup λ inducesa morphism

ϕ : K× → V

t )→ λ(t)v.

If we can extend ϕ to a morphism

ϕ : K → V



then we denote

ϕ(0) = limt→0

λ(t)v.

Remark 2.2.29. There is a slight abuse of notation in the definition as for the caseK = C,limt→0 λ(t)v also refers to the limit in the sense of analysis.. However, it is easy to seethat two definitions actually agree.

The Hilbert-Mumford Criterion (see [MFK94]), gives the connection between unstablepoints and one-parameter subgroups.

Theorem 2.2.30 (The Hilbert-Mumford Criterion I). Let G be a reductive group andV be a rational representation of V . A vector v ∈ V is unstable if and only if there existsa one-parameter subgroup λ : K× → G such that limt→0 λ(t)v = 0.

At the first glance, the Hilbert-Mumford criterion does not seem helpful as it replacesthe task of finding invariants with the task of finding all one-parameter subgroups.However, the following discussion shows that it is enough to consider the one-parametersubgroups of a fixed abelian subgroup of G.

Definition 2.2.31. A linear algebraic group G is said to be a torus if it is isomorphicto (K×)n for some n ∈ Z≥0. A maximal torus T of a linear algebraic group G is a closedsubgroup of G with the property that

1. T is a torus.

2. T is maximal among tori, i.e. T is not contained in any other closed subgroup ofG that is a torus.

Theorem 2.2.32. Let G be a reductive group. Given two maximal tori T, T ′ of G, thereexists g ∈ G such that gTg−1 = T ′, i.e. all maximal tori are conjugate in G.

Remark 2.2.33. The above theorem has the following consequence: Any one-parametersubgroup of G is a torus, hence contained in a maximal torus. Since all maximal toriare conjugate, the one-parameter subgroups considered in the Hilbert-Mumford criterioncan be replaced by the one-parameter subgroups of a fixed maximal torus T of G. Westate this fact formally in the following theorem:

Theorem 2.2.34 (The Hilbert-Mumford Criterion II). Let G be a reductive group, Vbe a rational representation and T ⊆ G be a maximal torus. Given v ∈ V , the followingare equivalent:

1. v is unstable.

2. There exists a one-parameter subgroup λ : K× → G such that limt→0 λ(t)v = 0.

3. There exists an element w ∈ G · v such that w ∈ NT (V ).

4. There exist a one-parameter subgroup λ : K× → T and g ∈ G such that

limt→0

λ(t)(g · v) = 0.



Example 2.2.35. As an application of the Hilbert-Mumford Criterion, we describe thenullcone of Matn,n under the action of GLn via conjugation. For all X ∈ Matn,n, thereexists g ∈ GLn such that Y = gXg−1 is an upper triangular matrices such that itsdiagonal entries are the eigenvalues of X. This is called the Jordan normal form of X.If all eigenvalues of X are 0, then X is strictly upper triangular. In this case, considerthe one-paramater subgroup

t )→ h(t) =

%

&&&'

tn−1

tn−2

. . .

1

(

)))*.

Observe that for any i, j we have (h(t)Y h(t)−1)ij = t(n−i)−(n−j)Yij = tj−iYij . SinceY is strictly upper triangular, for all i ≥ j we have Yij = 0. For i < j, the entryYij is multiplied with tj−i, which has positive exponent. Thus, h(t) annihilates Y , i.e.limt→0 h(t)Y h(t)−1 = 0. This shows that a matrix having all eigenvalues equal to 0 is inthe nullcone.

Conversely, assume that X ∈ Matn,n is in the nullcone. By the Hilbert-Mumfordcriterion, there exists g ∈ GLn and a one-parameter subgroup h(t) of the maximal torusof invertible diagonal matrices in GLn such that

limt→0

h(t)Y h(t)−1.

where Y = gXg−1. Write

h(t) =

%

&&&'

ta1

ta2

. . .

tan

(

)))*.

Observe that by applying a permutation matrix σ ∈ Sn ⊆ GLn to both h(t) and Y , wemay assume that a1 ≤ a2 ≤ · · · ≤ an. Then, for all i, j,

/h(t)Y h(t)−1)ij = tai−ajYij . For

i ≤ j, we have ai ≤ aj thus the exponent of tai−aj is not positive. As h(t) annihilatesY , we deduce that Yij = 0 for all i ≤ j. Thus, Y is strictly upper triangular. Thus, wededuce that a matrix X ∈ Matn,n is in the nullcone if and only if there exists g ∈ GLn

such that gXg−1 is strictly upper triangular. It is an easy exercise to show that this setactually is the set of all nilpotent matrices.

The Hilbert-Mumford criterion can be used to further analyze the structure of theaction via semi-invariants.

Definition 2.2.36. Let G be a reductive group, V be a rational representation and∆ = g ∈ G | g · v = v for all v ∈ V be the kernel of the representation. Assume thatσ : G → K× is a group character. A vector v ∈ V is called σ-semi-stable if there exists



a semi-invariant f ∈ K[V ]G,σ such that f(v) ∕= 0. Similarly, a vector v is called σ-stableif there exists a semi-invariant f ∈ K[V ]G,σ such that f(v) ∕= 0 and moreover, we havedimG · v = dimG/∆ and G · v is closed in the set w ∈ V | f(w) ∕= 0.

We note that given a one-parameter subgroup λ : K× → G and a group characterσ : G → K×, the composition

σ λ : K× → K×

is a morphism of linear algebraic groups. However, every morphism ϕ of linear algebraicgroups

ϕ : K× → K×

is of the form t )→ tn for some n ∈ Z. This leads to the following definition:

Definition 2.2.37. Given a one-parameter subgroup λ : K× → G and a group characterσ : G → K×, we define the pairing 〈λ,σ〉 ∈ Z to be the integer n where

σ λ : K× → K×

t )→ tn

The following theorem (see, for example, [Kin94], Proposition 2.5) classifies σ-semi-stable points using the Hilbert-Mumford Criterion:

Theorem 2.2.38 (The Hilbert-Mumford Criterion III). Let G be a reductive group andV be a rational representation. Moreover set

∆ = g ∈ G | g · v = v for all v ∈ V

to be the kernel of the action. Given a group character σ and a vector v ∈ V , v isσ-semi-stable if and only if

1. σ(∆) = 1

2. For every one-parameter subgroup λ : K× → G with the property that limt→0 λ(t)vexists, we have 〈λ,σ〉 ≤ 0.

A vector v ∈ V is σ-stable if and only if

1. v is σ-semi-stable

2. Every one-parameter subgroup λ : K× → G, for which limt→0 λ(t)v exists and〈λ,σ〉 = 0, satisfies λ(K×) ⊆ ∆.

2.2.5 Good Filtrations

Recall from the previous chapters that in positive characteristic, only few groups arelinearly reductive. However, we would still like to decompose rational representations ofreductive groups. Apparently, this is possible for a special class of rational representa-tions called the good modules. As it is impossible to present everything here, we referto Donkin’s book [Don85] for more information.



Definition 2.2.39. Let G be a reductive group. A Borel subgroup of G is a maximalclosed, connected, solvable subgroup of G.

Observe that since tori in G are closed, connected and abelian, every maximal torusof G is contained in a Borel subgroup of G.

Given a reductive group G, the representation theory of G is closely related to aconcept called a Borel pair.

Definition 2.2.40. Let G be a reductive group. A Borel pair of G is a pair (B, T ) of amaximal torus T of G and a Borel subgroup B of G containing T .

For each Borel pair (B, T ) of G, there is a canonical surjection ϕ : B → T .

Definition 2.2.41. Let G be a reductive group and (B, T ) a Borel pair. Let a groupcharacter λ : T → K× be given. Then we can extend λ to a group character λ : B → K×

via composing it with the canonical surjection ϕ : B → T . We define the dual Weylmodule ∇(λ) to be

∇(λ) = f ∈ K[G] | f(bg) = λ(b)f(g), ∀b ∈ B, g ∈ G.

Remark 2.2.42. Given a linear algebraic group G, a closed subgroup H and a rationalH-module V , the induced module IndGH(V ) is defined to be

IndGH(V ) = f : G → V | f is a morphism of varieties and ∀h ∈ H, g ∈ G, f(hg) = hf(g).

Observe that IndGH(V ) admits a natural G-module structure via

(g · f)(h) = f(g−1h).

With this notation, the dual Weyl module ∇(λ) is actually the induced module IndGB(K)where K is the one dimensional module on which B acts by the group character λ, i.e.b · v = λ(b)v for b ∈ B, v ∈ K.

Let T ∼= (K×)n be a torus. Every group character of (K×)n is of the form

ϕ : (K×)n → K×

(c1, . . . , cn) )→n!

i=1

caii

for some integer vector (a1, . . . , an) ∈ Zn. Thus, we can identify the group charactersof T with Zn. A group character λ = (λ1, . . . ,λn) of T is called dominant if λ1 ≥ λ2 ≥· · · ≥ λn.

Definition 2.2.43. We define a partial order on the set of group characters on T : Givenλ = (λ1, . . . ,λn), µ = (µ1, . . . , µn) ∈ Zn we write λ ≤ µ if

∀s ≤ n,

s+

i=1

λi ≤s+

i=1

µi



and,n

i=1 λi =,n

i=1 µi.

We call a set Π of dominant weights saturated if any dominant weight λ satisfyingλ ≤ µ for some µ ∈ Π belongs to Π.

Given a G-module V , we can restrict the action of G to a maximal torus T . As toriare linearly reductive, V decomposes as

V ="

λ

(Vλ)⊕mλ

where Vλ denotes the one-dimensional T -module where T acts via λ.

Theorem 2.2.44. Let G be a reductive group and V be an irreducible, rational G-module. Then there exists a dominant weight λ : T → K× such that

1. Vλ ∕= 0, i.e. λ appears as one of the weights of V

2. For any other weight µ such that Vµ ∕= 0, we have µ ≤ λ.

In this case we call λ the highest weight of V .

Proposition 2.2.45. The dual Weyl module ∇(λ) is finite dimensional. It is non-zeroif and only if λ is dominant. Moreover, let

L(λ) =+

U | U is an irreducible sub-module of ∇(λ)

be the socle of ∇(λ). Then the set

L(λ) | λ is a dominant weight

is a complete set of pairwise non-isomorphic irreducible G-modules.

Example 2.2.46. Assume that char(K) = 0 and G = GLn. Consider the Borel pair(Bn, Tn) where Bn is the set of invertible upper triangular matrices and Tn is the set ofinvertible diagonal matrices. Given a dominant weight (λ1, . . . ,λn) of Tn, the inducedgroup character is

(λ1, . . . ,λn) : B → K×%

&&&'

d1 ∗ . . . ∗0 d2 . . . ∗...

. . ....

0 . . . 0 dn

(

)))*)→

n!

i=1

dλii

If λ1 ≤ n and λn ≥ 0, then L(λ) is the irreducible GLn module Lλ′(Kn) where Lλ′ is theSchur functor corresponding to the partition λ′. If λ1 > n then L(λ) = Lλ′(Kn) = 0.

We can give the definition of a good module:



Definition 2.2.47. Let G be a reductive group and (B, T ) a fixed Borel pair. A G-module V is called a good G-module if there exists a filtration 0 = V0 ⊆ V1 ⊆ · · · ⊆ Vk =V of G-submodules of V such that for all i Vi/Vi−1 is either 0 or isomorphic to ∇(λ) forsome dominant group character λ of T . We call such a filtration a good filtration.

A subgroup H of G is called saturated if the induced module IndGH(K) is a goodG-module.

Example 2.2.48. Let T be a reductive group and consider G = T ×T × · · ·×T . Considerthe embedding T ∼= H = (g, g, . . . , g) | g ∈ G ⊆ G. Then H is a saturated subgroupof G. See [Don85] for the proof.

Definition 2.2.49. Let Π be a set of dominant weights on T . We say that V belongsto Π if there exists a filtration of G-submodules

0 = V0 ⊆ V1 ⊆ V2 ⊆ · · · ⊆ Vk = V

such that every factor Vi/Vi−1 is either 0 or isomorphic to L(λ) for some λ ∈ Π.

Theorem 2.2.50 (Ringel). Let F be the class of rational G-modules V such that both Vand V ∗ are good G-modules. For every dominant weight λ, there exists an indecomposableM(λ) in F . Moreover, every G-module V in F can be written as a direct sum of M(λ)’s.

Proposition 2.2.51. Assume that U and V are G-modules that belong to Π. ThenU ⊕ V also belong to Π.

Proof. Assume that0 = U0 ⊆ U1 ⊆ · · · ⊆ Ur = U

and0 = V0 ⊆ V1 ⊆ · · · ⊆ Vs = V

are the filtrations as in the above definition. Then

0 = U0 ⊕ 0 ⊆ U1 ⊕ 0 ⊆ · · · ⊆ Ur ⊕ 0 ⊆ Ur ⊕ V1 ⊆ · · · ⊆ Ur ⊕ Vs = U ⊕ V

has the property that each factor is either isomorphic to Vi/Vi−1 for some i or to Uj/Uj−1

for some j. Hence U ⊕ V belongs to Π.

Corollary 2.2.52. Every G-module V has a submodule OΠ(V ) that is maximal amongall submodules of V that belongs to Π.

Let G be a reductive group and H be a closed subgroup. Then H acts on G viaconjugation:

h · g = hgh−1

This action induces an action of H on K[G].

Definition 2.2.53. Given a reductive group G and a closed subgroup H, we defineC(G,H) to be the ring of invariants of the action of H on G via conjugation, i.e.

C(G,H) = f ∈ K[G] | f(hgh−1) = f(g), for all g ∈ G, h ∈ H.



Definition 2.2.54. Let G and H be an in the previous definition and let V be a rationalG-module with the corresponding representation ρ : G → GL(V ). Given a H-modulemap θ : V → V , we define the H-shifted trace function χθ by

χθ : G → K

g )→ Tr(θ ρ(g)).

Proposition 2.2.55. We have χθ ∈ C(G,H).

Proof.

χθ(hgh−1) = Tr(θρ(hgh−1)) = Tr(θρ(h)ρ(g)ρ(h−1)) = Tr(ρ(h)θρ(g)ρ(h)) = Tr(θρ(g))

Given a set Π of dominant weights, we define C(G,H,Π) to be the ring generated byinvariants of the form χθ where θ : V → V and V is a module that belongs to Π.

Theorem 2.2.56 ([Don92]). Let H be a closed, saturated subgroup of a reductive groupG and let Π be a saturated set of dominant weights. Then we have

OΠ(K[G])H = C(G,H,Π)

In particular, if Π is the set of all dominant weights then C(G,H) = C(G,H,Π).


3 Invariants and Semi-Invariants ofMatrices

A standard undergraduate linear algebra course starts by introducing the basic objects:vector spaces and linear maps. Then, it teaches students how to distinguish linear mapsfrom one another. More formally, given a vector space V , there is an induced vector spaceEnd(V ) which is the vector space of linear maps f : V → V . The group GL(V ) acts onEnd(V ) via

(g · f)(v) = gf(vg−1)

which amounts to a change of basis of V . A set of orbit representatives of this action isknown and called the Jordan normal forms. Using this theory, one can decide whethertwo linear maps f, g : V → V are in the same orbit or not.

Similarly, GL(V ) × GL(W ) acts on the vector space of linear maps f : V → W ,Hom(V,W ) via

((g, h) · f)(v) = hf(g−1v)

which amounts to a change of bases of both V and W . The normal forms for thisaction are again known and in fact much simpler than the Jordan normal form : Twolinear maps f, g : V → W are in the same orbit if and only if they have the samerank. In terms of invariant theory, this action is not very meaningful as there are nonon-constant invariants. Thus, we fix volume forms on V and W and then consider theaction of SL(V ) × SL(W ) on Hom(V,W ). Even though a bit more involved than thefirst one, normal forms are again known in this case.

As a general theme in invariant theory, we introduce the following natural generaliza-tions of these two actions:

1. The group GL(V ) acts on End(V )m via simultaneously conjugating each linearmap, i.e.

g · (f1, . . . , fm) = (g · f1, . . . , g · fm).

Here, g ·fi denotes the usual action of GL(V ) on End(V ) via conjugation as it is de-scribed above. This action is called simultaneous conjugation. Given (f1, . . . , fm)and (e1, . . . , em), we want to decide whether these two elements are in the sameorbit or not.

2. The group SL(V )× SL(W ) acts on Hom(V,W )m via simultaneously multiplyingeach linear map from left and right, i.e.

(g, h) · (f1, . . . , fm) = ((g, h) · f1, . . . , (g, h) · fm)


where (g, h) · fi denotes the usual action of SL(V )× SL(W ) on Hom(V,W ). Thisaction is called the left-right action. Given two tuples (f1, . . . , fm), (e1, . . . , em),we want to decide whether they are in the same orbit or not.

Even though these questions seem natural and simple, they are hard to tackle. In fact,there is no known set of normal forms for neither of these actions. Instead, we try toanswer the following easier(!) problems:

1. What are the polynomial functions F : End(V )m → K (where K is the base field)such that

F (f1, . . . , fm) = F (g · f1, . . . , g · fm) for all g ∈ GL(V )?

These polynomial functions are called matrix invariants in the literature.

2. What are the polynomial functions F : Hom(V,W )m → K such that

F (f1, . . . , fm) = F ((g, h) · f1, . . . , (g, h) · fm) for all g ∈ SL(V ), h ∈ SL(W )?

There polynomial functions are called matrix semi-invariants in the literature.

It is well-known that an answer to these two questions help us to give partial answersto our original problems : a description of invariants help us to distinguish the orbitclosures of the respective actions (but not the orbits). We also state the orbit closureintersection problem for these actions:

3. Given f = (f1, . . . , fm) and g = (g1, . . . , gm), decide whether O(f)∩O(g) is emptywhere O(f) denotes the orbit of f and O(f) denotes its Zariski closure.

When this problem is too hard to solve, we try to solve the a particular case of the orbitclosure problem that is called the nullcone membership problem:

4. Given f = (f1, . . . , fm), decide whether 0 ∈ O(f). In other words, decide whetherf is in the nullcone.

In this chapter, we give answers to all these problems for the mentioned actions. InSection 3.1, we describe the nullcone of End(V )m under simultaneous conjugation andwe introduce Procesi’s theorem that gives a set of generators of the ring of matrixinvariants. In Section 3.2, we start with an investigation of the nullcone of the left-right action via a concept called compression spaces. We give a set of generators forthe ring of matrix semi-invariants. In Section 3.3, we introduce degree bounds for aset of generators for the invariant ring for the considered actions. In Section 3.4, wegive efficient algorithms for the Orbit Closure Intersection Problem for both of theseactions.

Throughout this chapter, we fix an algebraically closed field K of arbitrary character-istic unless stated otherwise. We try to avoid making assumptions on the characteristicof the field as much as we can. However, even though it is possible to give a characteris-tic free description of a generating set of invariants for both of the actions, some of theknown degree bounds only work in 0 characteristic.



3.1 Matrix Invariants

The group GLn acts on the tuples of matrices via

g · (X1, X2, . . . , Xm) = (gX1g−1, . . . , gXmg−1)

where g ∈ GLn and (X1, . . . , Xm) ∈ Matmn,n.

We call this action simultaneous conjugation and denote its ring of invariants viaS(n,m) = K[Matmn,n]

GLn .

In [Art69] (p. 558), Artin conjectured that in characteristic 0, S(n,m) is generatedby traces of words in X1, . . . , Xm. Procesi proved this conjecture in [Pro76], and bystudying trace identities, Razmyslov ([Raz74]) proved that the invariants of degree ≤ n2

is enough to generate S(n,m).

In positive characteristic, Donkin [Don92] proved that it is enough to replace traceby coefficients of the characteristic polynomial of words in X1, . . . , Xm and in [DM16b],Derksen and Makam gave a polynomial bound in m and n for the degrees of the gener-ators.

3.1.1 A Description of the Nullcone of the Simultaneous Conjugation

We start by giving a description of the nullcone of the simultaneous conjugation. Thissection aims to convince the reader to the theorems of the following sections. The readermay skip this section as we do not use the following theorem in next sections.

Theorem 3.1.1. Let (X1, . . . , Xm) ∈ Matmn,n. Then the followings are equivalent:

1. (X1, . . . , Xm) is in the nullcone of the simultaneous conjugation.

2. There exists g ∈ GLn such that for all i, gXig−1 is a strictly upper triangular

matrix.

3. The (unital) subalgebra X generated by X1, . . . , Xm is nilpotent, i.e., every matrixX ∈ X is nilpotent.

Proof. (1 ⇒ 2): Suppose that (X1, . . . , Xm) is in the nullcone. By the Hilbert-Mumfordcriterion (see Theorem 2.2.34), there exists g ∈ GLn and an one-parameter subgroupλ : K× → Tn such that

limt→0

λ(t) · (gX1g−1, . . . , gXmg−1) = 0 (3.1)

where Tn is the maximal torus of GLn consisting of invertible diagonal matrices. Sinceg · X and X are in the same orbit, X is in the nullcone if and only if g · X is in thenullcone, thus we can replace X by g ·X.



As λ is an one-parameter subgroup of Tn, there exists (a1, . . . , an) ∈ Zn such that

λ : K× → Tn

t )→

%

&&&'

ta1

ta2

. . .

tan

(

)))*.

Without loss of generality, we may assume that a1 ≥ a2 ≤ · · · ≥ an as we can act on Xi

and on the one-parameter subgroup by a permutation matrix P ∈ Sn ⊆ GLn.

Then we have(λ(t)Xiλ(t)

−1)ij = tai−ajXij .

The exponent ai−aj is non-positive if i ≥ j. Using (3.1) above, we deduce that Xij = 0for all i ≥ j and the result follows.

(2 ⇒ 1): Let

λ(t) =

%

&&&'

tn−1

tn−2

. . .

1

(

)))*

Then for X = gXlg−1, we have (λ(t)Xλ(t)−1)ij = tj−iXij . As Xij = 0 for j − i ≤ 0,

every non-zero entry of Xij is multiplied by ta for some positive integer a. Thus, wehave

limt→0

λ(t) · (X1, . . . , Xm) = 0

which implies that (X1, . . . , Xm) is in the nullcone.

(2 ⇐⇒ 3) This equivalence is known as Engel’s Theorem (see, for example, [FH91],Theorem 9.9).

Remark 3.1.2. Here, the (unital) subalgebra generated by Xi is the smallest subspaceX of Matn,n that contains In and is closed under matrix multiplication. In other words,we have

X = 〈Xi1Xi2 . . . Xik | i1, i2, . . . , ik = 1, . . . ,m〉.

3.1.2 Generators of S(n,m) in Characteristic 0

In this section, we assume that char(K) = 0.

Given a finite set Σ, we define the Kleene closure Σ∗ as

Σ∗ = w1w2 . . . wn | wi ∈ Σ

i.e. the set of all words with alphabet Σ. We denote the empty word with ε.



To a matrix tuple (X1, . . . , Xm) ∈ Matmn,n and a word w = w1w2 . . . ws ∈ [m]∗ weassign a matrix

Xw = Xw1Xw2 . . . Xws

and a polynomial Trw(X1, . . . , Xm) = Tr(Xw). As a convention, Xε = In.

Lemma 3.1.3. For each w = w1 . . . ws ∈ [m]∗, we have Trw ∈ S(n,m) = K[Matmn,n]GLn.

Proof. For g ∈ GLn we have

(g · Trw)(X1, . . . , Xm) = Trw(g−1X1g, . . . , g

−1Xmg)

= Tr(g−1Xw1gg−1Xw2g . . . g

−1Xwsg)

= Tr(g−1Xwg)

= Tr(Xw)

= Trw(X1, . . . , Xm).

Procesi proved that the ring S(n,m) of matrix invariants is in fact spanned by poly-nomials of the form Trw for w ∈ [m]∗.

Theorem 3.1.4 ([Pro76]). Assume that char(K) = 0. Then S(n,m) is generated byTrw for w ∈ [m]∗.

Proof. It is clear by the previous lemma that Trw ∈ S(n,m) for each word w ∈ [m]∗.

Let f ∈ S(n,m) and recall from Section 2.1 the definition of a multilinear function.

Case 1: f is a multilinear function Matn,n × · · ·×Matn,n → K.

Let V = Kn. As in Section 2.1, we first identify multilinear functions Matn,n× · · ·×Matn,n → K with linear functions Matn,n⊗ · · · ⊗ Matn,n → K and using theidentification Matn,n ∼= V ∗ ⊗ V we assume that f is a linear function

f : (V ∗)⊗m ⊗ V ⊗m → K.

There is a natural pairing 〈·, ·〉 between Hom(V ⊗m, V ⊗m) and (V ∗)⊗m⊗V ⊗m. Onrank 1 tensors it is given by

〈·, ·〉 : Hom(V ⊗m, V ⊗m)× (V ∗)⊗m ⊗ V ⊗m → K

(ϕ,λ1 ⊗ · · ·⊗ λm ⊗ v1 ⊗ · · ·⊗ vm) )→ (λ1 ⊗ · · ·⊗ λm)(ϕ(v1 ⊗ · · ·⊗ vm)).

Here, (λ1 ⊗ · · ·⊗ λm)(v1 ⊗ · · ·⊗ vm) = λ1(v1) . . .λm(vm) is the usual evaluation offunctionals in (V ∗)⊗m ∼= (V ⊗m)∗.

Then, 〈·, ·〉 induces a GL(V ) isomorphism

π : Hom(V ⊗m, V ⊗m) → V ⊗m ⊗ (V ∗)⊗m

ϕ )→ 〈ϕ, ·〉



Also note that GL(V ) embeds into Hom(V ⊗m, V ⊗m) diagonally via g(v1 ⊗ · · · ⊗vm) = g(v1)⊗ · · ·⊗ g(vm).

Since f is a GL(V ) invariant, GL(V ) and π−1(f) commutes in Hom(V ⊗m, V ⊗m).

Now the Schur-Weyl duality (Theorem 2.1.22) implies that the centralizer ofGL(V )in the algebra Hom(V ⊗m, V ⊗m) is spanned by the elements of the form cσ whereσ ∈ Sm and

cσ(v1 ⊗ · · ·⊗ vm) = vσ−1(1) ⊗ · · ·⊗ vσ−1(m).

Thus, π−1(f) is an element of the subspace spanned by cσ. As a result, it is enoughto show the theorem for π(cσ).

Given σ ∈ Sm, denote π(cσ) by ψσ. Then we have

〈cσ,λ1 ⊗ . . .λm ⊗ v1 ⊗ · · ·⊗ vm〉 = (λ1 ⊗ · · ·⊗ λm)(cσ(v1 ⊗ · · ·⊗ vm))

= (λ1 ⊗ · · ·⊗ λm)(vσ−1(1) ⊗ · · ·⊗ vσ−1(m))

=!

i

λi(vσ−1(i))

= ψσ(λ1 ⊗ . . .λm ⊗ v1 ⊗ · · ·⊗ vm)

Recall that for rank 1 maps λ⊗ v and µ⊗ w with λ, µ ∈ V ∗, v, w ∈ V , we have

1. (λ⊗ v) · (µ⊗ w) = µ(v)λ⊗ w where · denotes the matrix product and

2. Tr(λ⊗ v) = λ(v).

If σ = (s1s2 . . . si1)(si1+1 . . . si2) . . . (sil+1 . . . sil+1) is the cycle decomposition of σ,

then we have

ψσ(λ1 ⊗ · · ·⊗λm ⊗ v1 ⊗ · · ·⊗ vm) =!

i

λi(vσ−1(i))

=!

i

λσ(i)(vi)

=#λs2(vs1)λs3(vs2) · · · λs1(vsi1 )

$· · ·

#λsil+2(vsil+1) · · · λsil+1(vsil+1

)$

(use 1 and 2) = Tr(

i1!

j=1

λsj ⊗ vsj ) · Tr(i2!

j=i1+1

λsj ⊗ vsj ) . . .Tr(

il+1!

j=sil+1

λsj ⊗ vsj )

= Tr(Xs1Xs2 . . . Xsi1) · · · Tr(Xsil+1 . . . Xsil+1

)

where Xi = λi ⊗ vi.

Hence ψσ assumes the expected form if the matrices are of rank 1: For rank1 matrices X1, . . . , Xm and σ = (s1s2 . . . si1)(si1+1 . . . si2) . . . (sil+1 . . . sil+1

) ∈ Sm,we have

ψσ(X1, . . . , Xm) = Tr(Xs1Xs2 . . . Xsi1) · · · Tr(Xsil+1 . . . Xsil+1

)



Both sides of the equation are multilinear functions Matn,n× · · · × Matn,n → Kand they agree on rank 1 matrices. Since rank 1 matrix tuples span Matmn,n, theymust agree everywhere. Thus, ψσ satisfies the result we wanted to show.

Case 2: f is an arbitrary homogeneous invariant of degree d.

Polarize f to obtain a multilinear function

F : Mat⊕mdn,n → K

with the property that

f(X1, . . . , Xm) = F (X1, . . . , Xm, X1, . . . , Xm, . . . , X1, . . . , Xm) (3.2)

where each matrix Xi is repeated d times. By the first step, F is in the span ofpolynomials

Trw, w ∈ [md]∗.

Observe that given w ∈ [md]∗,

Trw(X1, . . . , Xm, . . . , X1, . . . , Xm) = Trw′(X1, . . . , Xm)

for some w′ ∈ [m]∗ and we obtain the desired result.

Remark 3.1.5. Given a matrix X ∈ Matn,n, let σj be the j-th coefficient of the charac-teristic polynomial of X, i.e.

det(tIn +X) = tn + σ1tn−1 + σ2t

n−2 + · · ·+ σn−1t+ σn.

Observe that for g ∈ GLn we have

det(tIn + gXg−1) = det(g(tIn +X)g−1) = det(tIn +X)

so σj are in S(n, 1). In fact, S(n, 1) is generated by these polynomials. However, incharacteristic 0, the Newton identity gives

kσk =

k+

i=1

(−1)i−1σk−iTr(Xk)

which implies that σk ∈ K[Tr(Xi) | 1 ≤ i ≤ n] for all k = 1, . . . , n. The Newton identitydoes not hold in positive characteristic and the following example shows how the theoremfails in this case.

Example 3.1.6. Let K be the algebraic closure of F2. Then we have

Tr((I2)k) = Tr(I2) = 0



for all k but I2 is not in the nullcone of the simultaneous conjugation since gI2g−1 = I2

for all g ∈ GL2. Hence Tr(Xk) does not span the invariant ring S(n, 1) in positivecharacteristic. To that end, we need to replace Tr(Xk) with σk(X). For example wehave

S(2, 1) = K[σ1,σ2] = K[det,Tr]

in arbitrary characteristic.

Recall the First Fundamental Theorem of Invariant Theory for GL(V ) (see Theorem2.2.23): The invariant ring K[V p ⊕ (V ∗)q]GL(V ) is spanned by the polynomials 〈i | j〉where

〈i | j〉 : V p ⊕ (V ∗)q → K

(v1, . . . , vp,ϕ1, . . . ,ϕq) )→ ϕj(vi).

The full version of Procesi’s theorem can be stated as follows:

Theorem 3.1.7. The invariant ring K[Hom(V, V )m ⊕ V p ⊕ (V ∗)q] is spanned by thepolynomials of the form

1. Trw(X1, . . . , Xm) for w ∈ [m]∗ and

2. ϕj(Xwvi) for w ∈ [m]∗

where (X1, . . . , Xm, v1, . . . , vp,ϕ1, . . . ,ϕq) ∈ Hom(V, V )m ⊕ V p ⊕ (V ∗)q.

Proof. The ring K[Hom(V, V )m⊕V p⊕ (V ∗)q] naturally admits a multi-grading inducedby the given decomposition. Let f be an invariant and assume that the multi-degree off is (c1, . . . , cm, d1, . . . , dp, e1, . . . , eq). Moreover, set c =

,i ci, d =

,i di, e =

,i ei. If

f is non-zero, then d = e since for g = λ · idV we have

(g · f)(X1, . . . , Xm, v1, . . . , vp,ϕ1, . . . ,ϕq) = f(. . . , g−1Xig, . . . , g−1vj , . . . , g

−1ϕk, . . . )

= f(. . . , Xi, . . . ,λ−1vj , . . . ,λϕk, . . . )

= λe−df(. . . , Xi, . . . , vj , . . . ,ϕk, . . . ).

But since f is an invariant, we must have λe−d = 1 for all λ. As K is algebraicallyclosed, e = d.

We assume that f ∕= 0 so e = d. The polarization F of f is a multilinear function

F : Hom(V, V )c × V d × (V ∗)d → K

with the property that

f(X1, . . . , Xm, v1, . . . , vp,ϕ1, . . . ,ϕq) = F (X[ci]i , v

[ej ]j ,ϕ

[dk]k ). (3.3)

Here, by X[ci]i , we mean that Xi is repeated ci times and similarly for vj and ϕk. We have

F ∈ Hom(V, V )⊗c⊗ (V ∗)⊗d⊗V ⊗d. We note that if F satisfies the result of the theorem,



then so does f : If w ∈ [c]∗ is a word and we compute Xw for (X[ci]i | i = 1, . . . ,m), then

this matrix equals Xw′ for some w′ ∈ [m]∗. Thus, if F is in the span of the polynomialsgiven in the statement of the theorem, then so is f .

Consider the GL(V )-isomorphism

ψ : Hom(V, V )⊗c ⊗ (V ∗)⊗d ⊗ V ⊗d → Hom(V, V )⊗c+d

X1 ⊗ · · ·⊗Xc ⊗ ϕ1 ⊗ · · ·⊗ ϕd ⊗ v1 ⊗ · · ·⊗ vd )→ X1 ⊗ · · ·⊗Xc ⊗ (ϕ1 ⊗ v1)⊗ · · ·⊗ (ϕd ⊗ vd)

Since f is an invariant, ψ(F ) is a GL(V )-invariant in Hom(V, V )⊗c+d and by Procesi’stheorem, ψ(F ) can be written as a product of Trw for w ∈ [c + d]∗. Thus, it is enoughto show that each Trw pulls back under ψ to a product of the polynomials given in thestatement of the theorem.

Note that for a matrix X ∈ Hom(V, V ) and ϕ ∈ V ∗, v ∈ V , we have

X · (ϕ⊗ v) = ϕ⊗ (Xv) (3.4)

Here, the product on the left-hand side is matrix-matrix product and on the right-handside Xv denotes matrix-vector product.

Given w = [c+ d]∗, write

w = a1b1a2b

2 . . . akbk (3.5)

where ai ∈ [c]∗ and bi ∈ c + 1, c + 2, . . . , c + d∗. We might possibly have ai = ε, theempty word.

For a word b ∈ c+ 1, . . . , c+ d∗, let l(b) denote its length. Then we have

Trb(X1, . . . , Xc,ϕ1 ⊗ v1, . . . ,ϕd ⊗ vd) = Tr(

l(b)!

i=1

ϕbi ⊗ vbi)

=

3l(b)−1!

i=1

ϕbi+1(vbi)

4Tr(ϕb1 ⊗ vbl(b))

=

l(b)!

i=1

ϕbσ(i)(vbi)

where σ is the cycle (12 . . . l(b)) ∈ Sl(b).



Similarly, for a ∈ [c]∗ and b ∈ c+ 1, . . . , c+ d∗ we have

Trab(X1, . . . , Xc,ϕ1 ⊗ v1, . . . ,ϕd ⊗ vd) = Tr(Xa1Xa2 . . . Xal(a)ϕb1 ⊗ vb1 . . .ϕbl(b) ⊗ vbl(b))

=

3l(b)−1!

i=1

ϕbi+1(vbi)

4Tr(Xaϕb1 ⊗ vbl(b))

( use (3.4)) =

3l(b)−1!

i=1

ϕbi+1(vbi)

4Tr(ϕb1 ⊗ (Xavbl(b)))

=

3l(b)−1!

i=1

ϕbi+1(vbi)

4ϕb1(Xavbl(b))

Thus for w ∈ [c + d]∗ given as in (3.5), and X = (X1, . . . , Xc), v = (v1, . . . , vd),ϕ =(ϕ1, . . . ,ϕd) we have

(ψ−1Trw)(X, v,ϕ) = Trw(X1, . . . , Xc,ϕ1 ⊗ v1, . . . ,ϕd ⊗ vd)

= Tr(Xa1ϕb11⊗ vb11 . . .ϕb1

l(b1)⊗ vb1

l(b1). . .

Xakϕbk1⊗ vbk1

. . .ϕbkl(bk)

⊗ vbkl(bk)

)

=

3k!

i=1

l(bi)−1!

j=1

ϕbij+1(vbij

)

4Tr(ϕb11

(Xa1vb1l(b1)

) . . .ϕbk1(Xakvbk

l(bk)

)

=

3k!

i=1

l(bi)−1!

j=1

ϕbij+1(vbij

)

43k−1!

i=1

ϕbi+11

(Xaivbil(bi)

)

4Tr(ϕb11

⊗Xakvbkl(bk)

)

=

3k!

i=1

l(bi)−1!

j=1

ϕbij+1(vbij

)

43k−1!

i=1

ϕbi+11

(Xaivbil(bi)

)

4ϕb11

(Xakvbkl(bk)

)

Observe that each factor in this product is one of the forms given in the statementof the theorem and ψ−1Trw agrees with them on rank 1 tensors. As both sides aremultilinear, they must agree on the whole space. We deduce that F satisfies the resultof the theorem. On the other hand, f and F are related via (3.3) and thus f also satisfiesthe result.

Example 3.1.8. Let c = d = 2 and consider the word w = 1134214 ∈ [4]∗. Then we have

(ψ−1Trw)(X1, X2, v1, v2,ϕ1,ϕ2) = Tr#X1X1(ϕ1 ⊗ v1)(ϕ2 ⊗ v2)X2X1(ϕ2 ⊗ v2)

$

= ϕ2(v1) Tr#X1X1(ϕ1 ⊗ v2)X2X1(ϕ2 ⊗ v2)

$

= ϕ2(v1) Tr#(ϕ1 ⊗ (X1X1v2))(ϕ2 ⊗ (X2X1v2))

$

= ϕ2(v1)ϕ2(X1X1v2) Tr#ϕ1 ⊗ (X2X1v2)

$

= ϕ2(v1)ϕ1(X1X1v2)ϕ1(X2X1v2).



Moreover, if we set X1 = X2, v1 = v2 and ϕ1 = ϕ2 we get

f(X, v,ϕ) = (ψ−1Trw)(X,X, v, v,ϕ,ϕ) = ϕ(v)(ϕ(X2v))2

so f ∈ K[Hom(V, V )⊕ V ⊕ V ∗] assumes the expected form.

3.1.3 Generators of S(n,m) in Positive Characteristic

Now we will focus on the case char(K) > 0. Note that in Example 3.1.6, we showed thatProcesi’s theorem does not hold in positive characteristic. More precisely

K[Trw | w ∈ [m]∗]

is a proper subalgebra of the ring of invariants K[Matmn,n]GLn if char(K) > 0. In [Don92],

Donkin proved that if we also take the characteristic coefficients of words of matrices,then we can generate the ring of invariants.

Given w ∈ [m]∗ define the polynomial

σj,w : Matmn,n → K

(X1, . . . , Xm) )→ σj(Xw)

where σj denotes the j-th coefficient of the characteristic polynomial of Xw, i.e.

det(tIn +Xw) = tn + σ1tn−1 + · · ·+ σn−1t+ σn.

Proposition 3.1.9. For each j = 1, . . . , n and w ∈ [m]∗, the polynomial σj,w is a matrixinvariant.

Proof. Let g ∈ GLn be given. Then we have

det(tIn + gXwg−1) = det(g(tIn +Xw)g

−1) = det(tIn +Xw).

Thus σj(gXwg−1) = σj(Xw) and the result follows.

Donkin’s theorem [Don92] asserts that the invariant ring S(n,m) is generated by σj,w.The proof is involved with some techniques that we did not present. Therefore, we onlygive an outline of the proof. For the definitions, we refer to Section 2.2.5.

Recall the definition of a good filtration: Given a reductive group G, a maximal torusT and a rational module V of G, a good filtration is a filtration

0 = V0 ⊆ V1 ⊆ · · · ⊆ Vk = V

such that each factor Vi/Vi−1 is either 0 or isomorphic to the dual Weyl module ∇(λ)for some dominant group character λ of T . Moreover, we call a closed subgroup H of Gsaturated if the induced module

IndGH(K) = f ∈ K[G] | f(h · g) = f(g)

has a good filtration.



Proposition 3.1.10 ([Don88], Section 1, (5)). Let G = GLn × · · · × GLn where thereare m factors. Then GLn

∼= H = (g, . . . , g) | g ∈ GLn ⊆ G is a saturated subgroup ofG.

Let G and H be as above. There are 3 steps for the main theorem:

1. Find a saturated set Π of dominant weights of G such that OΠ(K[G]) = K[Matmn,n].

2. Show that every H-shifted trace function is spanned by σj,w.

3. Use Theorem 2.2.56 which states that OΠ(K[G])H is spanned by H-shifted tracefunctions.

We start with the first step. Assume that Π is a set containing some dominant weightsof G. For each λ ∈ Π, there exists an irreducible (if not zero) G-module L(λ) of G.Then, we say a G-module V belongs to Π if V admits a filtration

0 = V0 ⊆ V1 ⊆ · · · ⊆ Vk = V

such that each factor Vi/Vi−1 is either 0 or isomorphic to L(λ) for some λ ∈ Π. Observethat this condition is closed under taking direct sums, i.e. if both U and V belong toΠ, then so is their direct sum U ⊕ V . Thus, for each G-module V , there exists a uniquesubmodule OΠ(V ) that is the maximal submodule V that belongs to Π.

Lemma 3.1.11 ([Don87], Section 1, (1)). Let

Σ = (λ1, . . . ,λn) ∈ Zn | λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0

and let Π = Σ× Σ× . . .Σ be the m-fold product of Σ. Then we have

OΣ(K[GLn]) = K[Matn,n]

and hence

OΠ(K[G]) = K[Matmn,n].

Observe that if char(K) = 0, then the above lemma is the direct consequence of awell-known fact: If λ = (λ1, . . . ,λn) is a partition and Lλ is the irreducible GLn-modulewith highest weight vector λ, then Lλ is a polynomial representation of GLn if andonly if λn ≥ 0. In other words, OΣ(K[GLn]) contains every irreducible polynomialrepresentation of GLn.

By the above lemma, instead of computingK[Matmn,n]GLn , we can computeOΠ(K[G])H

where G = GLn × · · ·×GLn and H ∼= GLn is the diagonal subgroup of G, acting on Gby simultaneous conjugation. However, since Π is saturated set of dominant weights andthe subgroup H is saturated, this invariant ring is spanned by some special functionsthat are called H-shifted trace functions:



Definition 3.1.12. Let G be a reductive group, H be a closeed subgroup and V be arational G-module. Assume that ρ : G → GL(V ) is the corresponding representation.Given an H-module morphism θ : V → V , the H-shifted trace function χθ is defined tobe the map

χθ : G → K

g )→ Tr(θ ρ(g)).

Observe that χθ is invariant under the action of H on G via conjugation: For g ∈G, h ∈ H we have

χθ(hgh−1) = Tr(θ ρ(h) ρ(g) ρ(h)−1) = Tr

-ρ(h)

#θ ρ(g)

$ρ(h)−1

.= χθ(g).

Moreover, the invariant ringK[G]H is actually spanned byH-shifted trace functions.

Theorem 3.1.13 ([Don92], Theorem 1). Let G be a reductive group and H be a saturatedsubgroup of G. Let Π be a saturated set of dominant weights. Then, OΠ(K[G])H isspanned by H-shifted trace functions χθ where θ : V → V is an H-module map and Vis a rational G-module that belongs to Π.

Lemma 3.1.14 ([Don92] ). Let α(1),α(2), . . . ,α(k) ∈ Zm≥0. Set

V = Λα(1)(Kn)⊗ Λα(2)(Kn)⊗ · · ·⊗ Λα(k)(Kn)

= Λα(1)1(Kn)⊗ · · ·⊗ Λα(1)m(Kn)⊗ · · ·⊗ Λα(k)1(Kn)⊗ · · ·⊗ Λα(k)m(Kn).

Consider the diagonal action of G on V , i.e.

(g, g, . . . , g) · v1 ⊗ . . . vk = (gv1)⊗ · · ·⊗ (gvk)

where vi ∈ Λα(i)(Kn). Then for each H-module map θ : V → V , the H-shifted tracefunction χθ is in the space spanned by σj,w. Here, by σj,w we denote the restriction ofσj,w on G.

The importance of V in the above lemma is that each λ ∈ Π is the highest weight of Vfor some α(1),α(2), . . . ,α(k) ∈ Zm

≥0. Recall from Theorem 2.2.50 that for every dominantweight λ, there exists an indecomposable module M(λ) of G and every indecomposablemodule is of this form. Moreover, in [Don93], Donkin shows that V above contains thisindecomposable M(λ). Therefore, the above theorem holds for M(λ), and hence, forevery module that belongs to Π. We obtain the following theorem:

Theorem 3.1.15 (Donkin’s Theorem). The invariant ring K[Matmn,n]GLn is spanned by

σj,w for j = 1, . . . , n and w ∈ [m]∗.

In [Don92], Donkin also gives a generalization of the above theorem:



Theorem 3.1.16 ([Don92], Theorem 2). Let A ∈ Matn,n and let

CGLn(A) = g ∈ GLn | gA = Ag

be its centralizer in GLn. The invariant ring

K[Matmn,n]CGLn (A)

is spanned by the polynomials of the form

f : Matmn,n → K

(X1, . . . , Xm) )→ σj(Aq1Xw1A

q2Xw2 . . . AqkXwk

)

where w = w1w2 . . . wk ∈ [m]∗ and q1, . . . , qk ≥ 0.

Remark 3.1.17. Note that the given polynomial f is in fact an invariant of CGLn(A):For g ∈ CGLn(A) we have

f(gX1g−1, . . . , gXmg−1) = σj(A

q1gXw1g−1Aq2gXw2g

−1 . . . gXwkg−1)

= σj(gAq1Xw1gg

−1Aq2Xw2 . . . Xwk−1gg−1AqkXwk

g−1)

= σj(Aq1Xw1A

q2Xw2 . . . AqkXwk

)

= f(X1, . . . , Xm).

Example 3.1.18. The above theorem generalizes Donkin’s theorem since for A = In, wehave CGLn(A) = GLn.

Similarly, if λ1, . . . ,λn are distinct scalars in K and A is the diagonal matrix withentries λ1, . . . ,λn, then T = CGLn(A) is the set of invertible diagonal matrices in GLn.Assume that m = 1. If X ∈ Matn,n, then each diagonal entry of X is in fact an invariantof T . However, this invariant is not spanned by the characteristic coefficients of X. Toobtain these diagonal entries, we compute

Tr(Aq1X)

for distinct values of q1. These invariants are linear forms in the diagonal entries of X.Hence, for q1 = 0, 1, . . . , they span the linear space 〈X11, . . . , Xnn〉.

3.2 Matrix Semi-Invariants

Another group acting on matrix tuples is GLn ×GLn via

(g, h) · (X1, . . . , Xm) = (gX1h−1, . . . , gXmh−1)

for (X1, . . . , Xm) ∈ Matmn,n. We will call this action the left-right action.



Note that taking g = diag(t, t, . . . , t) for t ∈ K×, h = In and a homogeneous polynomialf ∈ K[Matmn,n] of degree d, we have

f(X1, . . . , Xm) = ((g, h)·f)(X1, . . . , Xm) = f(g−1X1h, . . . , g−1Xmh) = t−df(X1, . . . , Xm).

If d ∕= 0, then as K is algebraically closed, t−d ∕= 1 for some t ∈ K× which is acontradiction. Hence, the only invariants of GLn × GLn are constants. However, thespace of semi-invariants can still be interesting. Recall that in Definition 2.2.19, wedefined a semi-invariant of weight σ to be a homogeneous polynomial f such that g ·f =σ(g)f for all g ∈ G. Here, σ : G → K is a group character.

The following proposition is well-known:

Proposition 3.2.1. The group characters χ of GLn ×GLn are all of the form

χ(g, h) = det(g)k det(h)l

for some integers k, l ∈ Z. We will denote this group character by (k, l) if it does notlead to any confusion.

One consequence of this lemma is that the semi-invariants of GLn×GLn are the sameas the invariants of SLn × SLn.

We will denote the invariant ring K[Matmn,n]SLn×SLn with R(n,m). This ring is called

the ring of matrix semi-invariants. It is clear by the previous explanation that R(n,m)is the ring of semi-invariants for GLn ×GLn.

Example 3.2.2. For m = 1, and g, h ∈ SLn we have

det((g, h) ·X) = det(gXh−1) = det(g) det(h)−1 det(X) = det(X)

hence determinant is a matrix semi-invariant (of weight (−1, 1)). In fact, if det(X) = 0,then by multiplying from left, we can assume that the last row of X is 0 and then for

g(t) =

%

&&&'

tt

. . .

t1−n

(

)))*∈ SLn

we have limt→0 g(t)X = 0. We deduce that the nullcone of the left-right action for m = 1is the set of singular matrices (we actually have R(n, 1) = K[det]).

In this section, we will start with two descriptions of the nullcone of the left-rightaction and then give a set of generators of R(n,m).

3.2.1 Compression Spaces

Given a matrix X ∈ Matn,n, X is called singular if one of the equivalent conditionshold:



1. det(X) = 0.

2. There exists v ∈ Kn such that Xv = 0.

In Example 3.2.2, we showed that the nullcone of the action of SLn × SLn on Matn,n isthe set of singular matrices. This allows us to add one more equivalent condition to theabove list:

3. X is in the nullcone of the left-right action.

It turns out that it is possible to generalize this idea to tuples of matrices. In thissection, we will establish a generalization of the second condition of the above list totuples of matrices and show its equivalence with the third condition. Moreover, we givean explicit example of a tuple of matrices where the first condition holds but the secondand third conditions fail. We start with a definition:

Definition 3.2.3. A tuple (X1, . . . , Xm) ∈ Matmn,n is called a compression space if thereexist subspaces U, V ⊆ Kn such that for all i, Xi(U) ⊆ V and dimV < dimU .

If dimU − dimV = c then (U, V ) is called a c-shrunk subspace of (X1, . . . , Xm).

Remark 3.2.4. For m = 1, it should be clear that the above condition is equivalent to besingular: If det(X) = 0, then we can set U to be ker(X) and V = 0. Conversely, if Xcompresses a subspace U into V with dimV < dimU , then we can pick a complementW of U and obtain

dim(Im(X)) ≤ dim(X(U)) + dim(X(W )) ≤ dimV + n− dimU < n

so X does not have full rank. Consequently, if (X1, . . . , Xm) is a compression space,then for λ1, . . . ,λm ∈ K we have (λ1X1 + · · ·+ λmXm)(U) ⊆ V and hence

det(λ1X1 + · · ·+ λmXm) ≡ 0. (3.6)

In the special case that m = 2, the necessary condition (3.6) turns out to be suffi-cient:

Lemma 3.2.5. (X1, X2) ∈ Mat2n,n is a compression space if and only if det(sX1+tX2) ≡0 as a polynomial in s and t.

Proof. Only if part is clear by the previous remark.

For the if part, we will follow a proof by Burgin and Draisma (see [BD06] Theorem 2).

Assume that det(sX1 + tX2) ≡ 0. Consider the field K(s, t) of rational functions invariables s, t and the vector space V = K(s, t)n. Then X = sX1 + tX2 can be seen asan endomorphism of V . Since det(sX1 + tX2) ≡ 0, X is a singular endomorphism, so ithas a kernel 0 ∕= v ∈ V .



Write

v =

%

&&&'

f1/g1f2/g2

...fn/gn

(

)))*

where fi, gi ∈ K[s, t] and gi ∕= 0. Let g = g1g2 . . . gn. Then gv is also in the kernel of Xand each entry of gv is a polynomial. Replace v with gv.

We can decompose v as

v = sdw1 + sd−1tw2 + · · ·+ tdwk + lower degree terms

where wi ∈ Kn. Since each entry of sX1 + tX2 is a linear form in variables s, t, theequation

(sX1 + tX2)v = 0

implies that

(sX1 + tX2)(sdw1 + sd−1tw2 + · · ·+ tdwk) = 0

so we further assume that v = sdw1 + sd−1tw2 + · · ·+ tdwk.

If w1 or wk equals 0, then we can divide v by s or t, respectively. Hence, without lossof generality, w1 ∕= 0. The equation

0 = (sX1 + tX2)(sdw1 + · · ·+ tdwk)

= sd+1(X1w1) + sdt(X1w2 +X2w1) + · · ·+ std(X1wk +X2wk−1) + td+1(X2wk).

implies the following:

X1w1 = 0, X1w2 = −X2w1, X1w3 = −X2w2, . . . , X1wk = −X2wk−1, X2wk = 0.

Set U = 〈w1, . . . , wk〉. Then by the above equations, we getX1(U), X2(U) ⊆ 〈X1w1, . . . , X1wk〉.Since X1w1 = 0, dim〈X1w1, . . . , Xkwk〉 < dimU and the result follows.

Example 3.2.6. One may ask if det(λ1X1 + · · · + λmXm) ≡ 0 is enough to imply that(X1, . . . , Xm) is a compression space for all m. This is not the case. Consider

X1 =

%

'0 0 00 0 10 −1 0

(

* , X2 =

%

'0 0 10 0 0−1 0 0

(

* , X3 =

%

'0 1 0−1 0 00 0 0

(

*

Then each linear combination of X1, X2, X3 is a skew-symmetric matrix and sincen = 3, its determinant is necessarily zero. However, a direct computation shows that(X1, X2, X3) is not a compression space: For any non-zero vector u = (x, y, z) ∈ K3, wehave

X1u = (0, z,−y), X2u = (z, 0,−x), X3u = (y,−x, 0).



Consider the matrix %

'X1uX2uX3u

(

* =

%

'0 z −yz 0 −xy −x 0

(

* .

For any 0 ∕= u ∈ K3, this matrix has rank 2: Its determinant is always 0 but x2, y2, z2

appears as 2 × 2-minors of the matrix which implies that if (x, y, z) ∕= 0 then it has anon-zero 2 × 2-minor. As a result, (X1, X2, X3) maps every 1-dimensional space to a2-dimensional space. Thus, (X1, X2, X3) cannot compress U if dimU = 1 or 2. On theother hand, if dimU = 3 and V = X1(U) +X2(U) +X3(U) has dimension less than 3,then there exists a non-zero common vector in the kernels of XT

1 , XT2 , X

T3 . This is also

not the case as XTi = −Xi and the above explanation also implies that −X1,−X2,−X3

maps every line to a plane.

The example shows that the conditions given in the beginning of the section are nolonger equivalent for m ≥ 3. However, it turns out that being a compression space isequivalent to be in the nullcone of the left-right action.

Theorem 3.2.7 ([BD06]). Consider the left-right action of SLp × SLq on the vectorspace Matmp,q. A tuple (X1, . . . , Xm) is in the nullcone of this action if and only if thereexist subspaces U ⊆ Kq, V ⊆ Kp subject to the conditions that

• for all i, we have Xi(U) ⊆ V and

• p · dimU > q · dimV .

Proof. (⇒) Assume that (X1, . . . , Xm) is in the nullcone of the left-right action. Thenby the Hilbert-Mumford criterion (see Theorem 2.2.30), there exists a one-parametersubgroup (λ, µ) : K× → SLp × SLq that annihilates (X1, . . . , Xm), i.e.

∀i, limt→0

λ(t)Xiµ(t)−1 = 0

Without loss of generality (see Remark 2.2.33), we may assume that λ(t) and µ(t) arediagonal, so we can write

λ(t) =

%

&&&'

ta1 0 . . . 00 ta2 . . . 0...

.... . .

...0 0 . . . tap

(

)))*, µ(t) =

%

&&&'

tb1 0 . . . 00 tb2 . . . 0...

.... . .

...0 0 . . . tbq

(

)))*

Since λ(t) ∈ SLp, µ(t) ∈ SLq, we have,

ai =,

bi = 0.

Recall that a flow network N is a directed graph with two distinguished vertices s, t,called the source and the sink respectively and a capacity function c that assigns eachedge a non-negative real number or ∞. A (feasible) flow is a function f that assigns each



edge a non-negative real number that does not exceed the capacity with the additionalcondition that

∀v ∈ V (N), s ∕= v ∕= t,+

w→v

f(w → v) =+

v→w

f(v → w),

i.e. the flow entering a vertex v equals to the flow leaving v. We call this conditionconservation of flow. The value of the flow f , denoted by |f |, is the net flow entering t.By conservation of flow, this value also equals to the net flow leaving s.

A cut of a flow network N is a partition (S, T ) of vertices such that s ∈ S and t ∈ T .The value of a cut (S, T ) is the sum of the capacities of all edges v → w with v ∈ S andw ∈ T . The max-flow min-cut theorem states that the maximum value of a flow equalsto the minimum value of a cut. For the precise definitions and the theorem, see [Wes00],Chapter 4 Section 3.

Let v1, . . . , vp be the standard basis vectors of Kp, and let w1, . . . , wq be the standardbasis vectors of Kq. Consider the network with vertex set s, t, v1, . . . , vp, w1, . . . , wq.For each i = 1, . . . , p, draw an (directed) edge from s to vi capacity q and for eachi = 1, . . . , q, draw an edge from wi to t of capacity p. Also, draw an edge of infinitecapacity from vi to wj if ai − bj > 0.

Since 0 is contained in the closure of (λ, µ) · (X1, . . . , Xm), for each i, j with ai−bj ≤ 0we have Xij = 0. We deduce the following:

∀k = 1, . . . ,m ∀i = 1, . . . , p Xk(vi) ∈ 〈wj | there exists an edge vi → wj〉. (3.7)

Observe that since there are p edges emanating from s with capacity q, any flow ofthis network has capacity ≤ pq. We will show that there is no flow with |f | = pq.

To reach a contradiction, suppose that |f | = pq and fij is the value of this flow on theedge vi → wj . Extend f on all pairs of vertices (vi, wj) by assigning fij = 0 if there is noedge from vi to wj . Then, by conservation of flow, we have

,j fij = q and

,j fji = p.

Thus, +

i,j

fij(bj − ai) = p+

j

bj − q+

i

ai = 0.

However, there is an edge vi → wj in the network if and only if bj − ai > 0. Thus, sincef is not zero on all edges of the network, the left-hand side of this equation is strictlypositive. We get a contradiction.

Using max-flow min-cut theorem, there is a cut (S, S) of capacity < pq. Let

U = vi | vi ∈ S, V = wj | wj ∈ S

i.e. U is the set of vertices that are not cut off from s and V is the set of vertices thatare cut off from t. Since the capacity of this cut is smaller than pq, in particular there



is no edge of infinite capacity in the cut, i.e. there is no edge vi → wj with vi ∈ U andwj ∕∈ V . Moreover, the capacity of this cut equals

q|U |+ p(q − |V |) < pq

Set U = 〈vi | vi ∈ U〉 ⊆ Kp and V = 〈wj | wj ∈ V 〉 ⊆ Kq. By the (3.7) we haveXi(U) ⊆ V for all i and q dim U +p(q−dim V ) < pq so p dim U > q dim V . We obtainedthe desired result.

(⇐) Let U ⊆ Kq, V ⊆ Kp be as in the assumption. Let B = v1, . . . , vq be a basis ofKq such that the last dimU vectors span U . Similarly, let B′ = w1, . . . , wp be a basisof Kp such that the last dimV vectors span V .

Let P be the transition matrix from B′ to the standard basis of Kp and Q be thetransition matrix from B to the standard basis of Kq. Set g = P

det(P ) and h = Qdet(Q) .

Then for all i we have

gXih−1 =

%

&&&&&&&&'

∗ . . . ∗ ∗ . . . ∗...

. . ....

.... . .

...∗ . . . ∗ ∗ . . . ∗∗ . . . ∗ 0 . . . 0...

. . ....

.... . .

...∗ . . . ∗ 0 . . . 0

(

))))))))*

where the size of the 0 block is (p− dimV )× dimU . Replace

(X1, . . . , Xm) ← (gX1h−1, . . . , gXmh−1)

and consider the following one-parameter subgroup of SLp

λ(t) =

%

&&&&&&&&'

tq(p−dimV )

. . .

tq(p−dimV )

0

0t−q dimV

. . .

t−q dimV

(

))))))))*

dimV

p−dimV

and of SLq

µ(t) =

%

&&&&&&&&'

t−p dimU

. . .

t−p dimU

0

0tp(q−dimU)

. . .

tp(q−dimU)

(

))))))))*

q−dimU

dimU



The corresponding one-parameter subgroup (λ(t), µ(t)) ∈ SLp×SLq annihilates Xi since

λ(t)Xiµ(t)−1 =

%

&&&&&&'

tpq+p dimU−q dimV ·∗ tp dimU−q dimV ·∗

tp dimU−q dimV ·∗ 0

(

))))))*

and by assumption p dimU − q dimV > 0.

Corollary 3.2.8. A tuple (X1, . . . , Xm) ∈ Matmn,n is in the nullcone of the left-rightaction if and only if (X1, . . . , Xm) is a compression space.

Remark 3.2.9. In the above corollary, we use Theorem 3.2.7 by taking p = q = n. On theother hand, this case can be proven more easily by replacing max-flow min-cut theoremin the proof of Theorem 3.2.7 by a weaker version of it that is Hall’s marriage theorem.

Definition 3.2.10. Let X ⊆ Matn,n. A c-shrunk subspace of X is a pair (U, V ) ofsubspaces of Kn such that X (U) ⊆ V and dimU − dimV = c.

With this new definition, it is apparent that a tuple (X1, . . . , Xm) is a compressionspace if and only if X = 〈X1, . . . , Xm〉 has a c-shrunk subspace for some c ∈ Z≥1.

The existence of c-shrunk subspace can be tested using a concept called the secondWong sequence:

Definition 3.2.11. Let X ⊆ Matn,n be a subspace and A ∈ X . The second Wongsequence of (A,X ) is a sequence W0 ⊆ W1 ⊆ . . . of subspaces of Kn given by

W0 = 0, W1 = X (A−1(W0)), . . . , Wi = X (A−1(Wi−1)), . . .

Proposition 3.2.12. There exists 0 ≤ l ≤ n such that

W0 ⊆ W1 ⊆ · · · ⊆ Wl = Wl+1 = Wl+2 = . . .

In this case, we call Wl the limit of the second Wong sequence and denote it by W ∗.Alternatively, W ∗ is the smallest subspace of Kn satisfying

X (A−1(W ∗)) = W ∗.

Proof. Observe that if Wi = Wi+1 then Wi+1 = Wi+2 as

Wi+2 = X (A−1(Wi+1)) = X (A−1(Wi)) = Wi+1.

This fact is enough to show that the sequence stabilizes: Since Wj ⊆ Wj+1, dimWj isan increasing sequence and since Wi ⊆ Kn, it stabilizes for some l ≤ n.



For the second claim, assume that V ⊆ Kn satisfies X (A−1(V )) = V . If there existsan index i such that Wi ⊆ V , then we have

Wi+1 = X (A−1(Wi)) ⊆ X (A−1(V )) = V

so Wi+1 is also contained in V . As W0 = 0 ⊆ V , we deduce that V contains the secondWong sequence and in particular W ∗ ⊆ V . We obtained the desired result.

Lemma 3.2.13 ([IQKS13],[FR]). Let A ∈ X and W ∗ be the limit of the second Wongsequence of (A,X ). There exists a (n − rk(A))-shrunk subspace of X if and only ifW ∗ ⊆ Im(A). In this case, U = A−1(W ∗) and V = X (U) form the shrunk subspace.

Proof. (⇒) Let (U, V ) be a n− rk(A)-shrunk subspace, i.e.

X (U) ⊆ V, dimU − dimV = n− rk(A).

As A ∈ X , we have A(U) ⊆ V so dimU −dimA(U) ≥ n− rk(A). The reverse inequalityholds for any matrix A and subspace U , so we deduce

dimU − dimA(U) = n− rk(A).

This equality implies that ker(A) ⊆ U so A−1(A(U)) = U .

As A(U) ⊆ X (U), we obtain dimU −dimX (U) ≤ dimU −dimA(U) = n− rk(A). Onthe other hand, we have dimU − dimX (U) ≥ dimU − dimV = n− rk(A). Thus

dimU − dimX (U) = n− rk(A).

Combining two equalities, we obtain dimA(U) = dimX (U) and hence, A(U) = X (U).Then, we have

A−1(X (U)) = A−1(A(U)) = U

and X (A−1(X (U))) = X (U). Using the above proposition, we get W ∗ ⊆ X (U) = A(U)which implies that W ∗ ⊆ Im(A).

(⇐) Set U = A−1(W ∗) and V = X (U). As W ∗ ⊆ Im(A), we have

dimU = dimW ∗ + dimker(A) = dimW ∗ + n− rk(A).

On the other hand, we have V = X (U) = X (A−1(W ∗)) = W ∗ thus dimV = dimW ∗.Combining these equalities, we obtain

dimU − dimV = dimW ∗ + n− rk(A)− dimW ∗ = n− rk(A)

so (U, V ) form a n− rk(A)-shrunk subspace.

In [IQKS13], the authors describe a polynomial time algorithm to detect whetherW ∗ ⊆ Im(A). We will discuss this algorithm in Section 3.4.



3.2.2 Tensor Blow-Ups

In the previous section, we have described the nullcone of the left-right action on matrixtuples via compression spaces. In this section, we will discuss a concept called tensorblow-ups which will help us to give a complete set of generators of R(n,m) in the nextsection.

The main theorems of this section are the regularity lemma of tensor blow-ups byIvanyos, Qiao and Subrahmanyam ([IQS15b]) and the concavity of the rank function byDerksen and Makam [DM15].

Recall that given two matrices X ∈ Mata,b and Y ∈ Matp,q, their tensor product (orKronecker product) is defined to be

X ⊗ Y =

%

&&&&'

X11Y X12Y . . . X1bY

X21Y. . .

......

. . ....

Xa1Y . . . . . . XabY

(

))))*∈ Matap,bq .

Definition 3.2.14. Given a subspace X = 〈X1, . . . , Xm〉 ⊆ Matn,n, its (p, q)-tensorblow-up is defined to be

X p,q = X ⊗Matp,q = 〈X ⊗ T | X ∈ X , T ∈ Matp,q〉

= m+

i=1

Xi ⊗ Ti | Ti ∈ Matp,q ⊆ Matnp,nq

We abbreviate X d,d by X d and call this space d-th tensor blow-up of X .

For a subspace X ⊆ Matp,q, define its rank to be

rk(X ) = maxrk(X) | X ∈ X.

Then it is clear for a subspace X ⊆ Matn,n that

rk(X d) ≥ rk(X )d

since for a matrix X ∈ X of rank rk(X ) we have

X ⊗ Id ∈ X d, rk(X ⊗ Id) = d rk(X) = d rk(X ).

At the first glance, one may think that rk(X d) = d rk(X ) holds. However, the followingexample by Domokos [Dom00] shows that this is not the case.

Example 3.2.15. Let

X =

5X1 =

%

'0 0 00 0 10 −1 0

(

* , X2 =

%

'0 0 10 0 0−1 0 0

(

* , X3 =

%

'0 1 0−1 0 00 0 0

(

*6

⊆ Mat3,3



be the subspace of skew-symmetric matrices. Since X consists of skew-symmetric ma-trices and every skew-symmetric 3× 3-matrix is singular, we have rk(X ) = 2. However

X1 ⊗70 00 1

8+X2 ⊗

70 11 0

8+X3 ⊗

71 00 0

8=

%

&&&&&&'

0 0 1 0 0 10 0 0 0 1 0−1 0 0 0 0 00 0 0 0 0 10 −1 0 0 0 0−1 0 0 −1 0 0

(

))))))*(3.8)

is invertible so rk(X 2) = 6.

The following lemma shows a surprising property of tensor blow-ups. It is called theregularity lemma for tensor blow-ups and crucial for the study of R(n,m).

Lemma 3.2.16 (The Regularity Lemma). Let X ⊆ Matn,n be a subspace. Thenrk(X d) is divisible by d.

We postpone the proof of this lemma to Section 3.4 where we prove a constructiveversion of this lemma by Ivanyos, Qiao and Subrahmanyam (see [IQS15b], [IQS15a]).Instead, we continue analyzing the properties of rk(X p,q).

Definition 3.2.17. Let X ⊆ Matn,n be a subspace. Define the function r : Z≥0×Z≥0 →Z≥0 by

r(p, q) = rk(X p,q)

Lemma 3.2.18 ([DM15]). Fix a subspace X ⊆ Matn,n. The function r has the followingproperties:

(1) r(p, q + 1) ≥ r(p, q),

(2) r(p+ 1, q) ≥ r(p, q),

(3) r(p, q + 1) ≥ 12(r(p, q) + r(p, q + 2)),

(4) r(p+ 1, q) ≥ 12(r(p, q) + r(p, q + 2)),

(5) r(p, q) is divisible by gcd(p, q).

Proof. (1): Consider the embedding

ϕ : Matp,q → Matp,q+11c1 c2 . . . cq

2)→

1c1 c2 . . . cq 0

2

where ci denotes the i-th column of the matrix. Then ϕ induces a rank preservingembedding of X ⊗Matp,q into X ⊗Matp,q+1. Hence r(p, q) ≤ r(p, q + 1) and (2) followsby symmetry.



(3): Let T = (T1, . . . , Tm) ∈ Matmp,q+2. For a subset J ⊆ 1, . . . , q + 2, let T Ji be the

submatrix of Ti where all the columns with index in J are omitted, and let YJ be thecolumn span of

,iXi ⊗ T J

i . Now, observe that the condition

rk(+

i

Xi ⊗ Ti) < r(p, q + 2)

is Zariski closed as it is given by r(p, q + 2)-minors of,

iXi ⊗ Ti. Hence for a generalenough T = (T1, . . . , Tm), we have

∀J rk(+

i

Xi ⊗ T Ji ) = r(p, q + 2− |J |)

Note that

Y1 + Y2 = Y∅ and Y1,2 ⊆ Y1 ∩ Y2

Thus

r(p, q) = dimY1,2 ≤ dimY1∩Y2 = dimY1+dimY2−dim(Y1+Y2) = 2r(p, q+1)−r(p, q+2)

and the result follows. (4) follows by symmetry.

(5): Let d = gcd(p, q) and write p = dp′, q = dq′. Then

X p,q = (X p′,q′)d

and the result follows by Lemma 3.2.16.

Corollary 3.2.19. r(p, q) is weakly increasing and weakly concave in either variable,i.e. for b > a we have

r(p, q + a) ≥ 1

b(ar(p, q + b) + (b− a)r(p, q)).

Furthermore, we have:

Lemma 3.2.20 ([DM15]). If r(1, 1) = 1 then we have r(d, d) = d for all d.

Proof. Observe that X ⊗Mat1,1 is isomorphic to X so 1 = r(1, 1) = rk(X ). Let 0 ∕= A ∈X . Since rk(A) = 1, there exist g, h ∈ GLn such that

gAh−1 =

%

&&&'

1 0 . . . 00 0 0...

. . ....

0 0 . . . 0

(

)))*



Replace X with gXh−1. Now for B ∈ X , if Bij ∕= 0 for some i > 1, j > 1, then we have

tA+B =

%

&&&&&&'

t+B11 . . . B1j . . . B1n...

. . ....

Bi1 Bij...

. . ....

Bn1 . . . . . . . . . Bnn

(

))))))*

so tBij +B11Bij −Bi1B1j is a 2-minor of tA+B. Observe that since Bij ∕= 0, this minoris non-zero for some values of t, which contradicts the fact that rk(X ) = 1. Hence

X ⊆

%

&&&'

∗ ∗ . . . ∗∗ 0 . . . 0...

.... . .

...∗ 0 . . . 0

(

)))*

However since every matrix 0 ∕= B ∈ X has rank 1, B1j and Bi1 cannot be both nonzerofor i > 1, j > 1. Using the fact that X is a subspace, we get

X ⊆

%

&&&'

∗ ∗ . . . ∗0 0 . . . 0...

.... . .

...0 0 . . . 0

(

)))*, or X ⊆

%

&&&'

∗ 0 . . . 0∗ 0 . . . 0...

.... . .

...∗ 0 . . . 0

(

)))*

Thus, every matrix in X ⊗ Matd,d has at most d non-zero rows or at most d non-zerocolumns which implies that r(d, d) ≤ d. Since A ⊗ Id has rank d, we have r(1, 1) ≥ dand the result follows.

Remark 3.2.21. The proof of the above lemma shows an interesting fact: If rk(X ) = 1,then by multiplying from left and right, we can assume that all matrices in X has onlyone non-zero column or only one non-zero row. When X = 〈X〉 is one-dimensional, thisfact is obvious. If rk(X) = r, then we can apply a row or a column change and assumethat X has only r non-zero rows or r non-zero columns. This no longer holds if X is notone-dimensional and shows how the theorems from classical linear algebra may fail fortuples of matrices.

Lemma 3.2.22 ([DM15]). Let n ≥ 2 and let d + 1 ≥ n. If r(d + 1, d + 1) = n(d + 1),then r(d, d) = nd as well.

Proof. Say r(d+ 1, d+ 1) = n(d+ 1). For 1 ≤ a ≤ d, the weak concavity implies that

r(d+ 1, a) ≥ (d+ 1− a)r(d+ 1, 0) + ar(d+ 1, d+ 1)

d+ 1=

an(d+ 1)

d+ 1= an

Conversely, r(d + 1, a) ≤ an is clear so r(d + 1, a) = an. By symmetry we also haver(a, d + 1) = an. Note that r(1, 1) ∕= 1 since otherwise by the previous lemma we have



r(d+ 1, d+ 1) = d+ 1 ∕= n(d+ 1). Hence r(1, 1) ≥ 2 and we have

r(1, d) ≥ (d− 1)r(1, d+ 1) + r(1, 1)

d≥ (d− 1)n+ 2

d= n− n− 2

d> n− 1

Since r(1, d) is an integer, r(1, d) ≥ n. By weak concavity in the first variable, we get

r(d, d) ≥ (d− 1)r(d+ 1, d) + r(1, d)

d≥ (d− 1)nd+ n

d= nd− n+

n

d

Since d ≥ n− 1, nd−n+ nd > d(n− 1) so r(d, d) > d(n− 1). However by Lemma 3.2.16,

r(d, d) is divisible by d so r(d, d) ≥ dn. The inequality r(d, d) ≤ nd is clear so the resultfollows.

3.2.3 Generators of R(n,m)

In this section, we give a set of generators for the invariant ring R(n,m).

Recall that for m = 1, we have R(n, 1) = K[det] and for m = 2, the invariantring is generated by the coefficients of the polynomial det(sX1 + tX2). For n,m ≥ 3,the coefficients of det(t1X1 + · · · + tmXm) are invariants but they do not generate theinvariant ring. An example of this fact is Example 3.2.6. However, there still is adeterminantal description of invariants which depends on the ideas introduced in theprevious section.

Proposition 3.2.23. Let (X1, . . . , Xm) ∈ Matmn,n and T = (T1, . . . , Tm) ∈ Matmd,d begiven. Then

fT (X1, . . . , Xm) = det(X1 ⊗ T1 + · · ·+Xm ⊗ Tm)

is a matrix semi-invariant, i.e. it is an invariant under the action of SLn × SLn onMatmn,n via the left-right action.

Proof. Let g, h ∈ SLn be given. Then we have

((g, h) · fT )(X1, . . . , Xm) = fT (g−1X1h, . . . , g

−1Xmh)

= det((g−1X1h)⊗ T1 + · · ·+ (g−1Xmh)⊗ Tm)

= det((g−1 ⊗ Id)(X1 ⊗ T1 + · · ·+Xm ⊗ Tm)(h⊗ Id))

= det(g−1 ⊗ Id) det(h⊗ Id) det(X1 ⊗ T1 + · · ·+Xm ⊗ Tm)

= det(X1 ⊗ T1 + · · ·+Xm ⊗ Tm)

= fT (X1, . . . , Xm).

Remark 3.2.24. For A ∈ Matp,p and B ∈ Matq,q we have

det(A⊗B) = det(A)q det(B)p



so using the above proof one can easily see that fT is a GLn × GLn semi-invariantcorresponding of weight

χ(−d,d) : GLn ×GLn → K

(g, h) )→ det(g)−d det(h)d.

In fact, no other group character appears as a weight of a non-constant homogeneoussemi-invariant: If f ∈ K[Matmn,n] is a homogeneous semi-invariant of degree k ≥ 1 andhas weight (a, b) ∈ Z2, then for

g(t) =

%

&&&'

tt

. . .

t

(

)))*, h(t) =

%

&&&'

tt

. . .

t

(

)))*

we have(g(t), h(t)) · f = tn(a+b)f, for all t ∈ K×

which implies for all t ∈ K× that

tn(a+b)f(X1, . . . , Xm) = ((g(t), h(t)) · f)(X1, . . . , Xm)

= f(g(t)−1X1h(t), . . . , g(t)−1Xmh(t))

= f(X1, . . . , Xm).

If a+ b ∕= 0, this equality does not hold for at least one t ∈ K× since K is algebraicallyclosed. Thus, a+ b = 0. Moreover, a ≤ 0 since for

g(t) =

%

&&&'

tt

. . .

t

(

)))*

and h = In we have

tnaf(X1, . . . , Xm) = ((g(t), In)·f)(X1, . . . , Xm) = f(t−1X1, . . . , t−1Xm) = t−kf(X1, . . . , Xm)

which implies that na = −k.

Remark 3.2.25. Let

X1 =

%

'0 0 00 0 10 −1 0

(

* , X2 =

%

'0 0 10 0 0−1 0 0

(

* , X3 =

%

'0 1 0−1 0 00 0 0

(

* .

In Example 3.2.15, we have shown that there exists T1, T2, T3 such that

det(X1 ⊗ T1 +X2 ⊗ T2 +X3 ⊗ T3) ∕= 0.

Thus, the above proposition gives us another criterion which shows that (X1, X2, X3) isnot in the nullcone of the left-right action.



It next theorem proves that the invariants fT , T ∈ Matmd,d actually span the ring ofinvariants. This theorem is usually given as the corollary of a more general theorem thatdescribes a set of generators for the ring of semi-invariants of quiver representations. Wewill introduce this concept in the next chapter. To keep this chapter self-contained, weadapted the proof so that we do not use the language of quiver representations.

Theorem 3.2.26. The subspace of matrix semi-invariants of weight (−d, d) is linearlyspanned by fT where T ∈ Matmd,d.

Proof. We prove the theorem in three steps.

In the first step, we will show that if we define the map

π : Matmn,n → Matmd2

nd,nd

(X1, . . . , Xm) )→ (Ejk ⊗Xi | i ∈ [m], j, k ∈ [d]),

then π( is a surjection from the semi-invariants of Matmd2

nd,nd of weight (−1, 1) (under theaction of GLnd × GLnd) to the semi-invariants of Matmn,n of weight (−d, d). Since thegroups acting on these spaces are different, this is not easy to show directly. Thus, wewill define two maps

ψ : Matmn,n → U

ϕ : U → Matmd2

nd,nd

where U is a GLnd×GLn×GLn×GLnd-module. We will show that ϕ( gives a surjectionfrom the semi-invariants of Matmd2

nd,nd of weight (−d, d) to the semi-invariants of U of some

certain weight. Similarly, we will show that this space surjects via ψ( onto the semi-invariants of Matmn,n of weight (−d, d).

In the second step, we show that the composition of the two maps above is actuallyπ and the semi-invariants of the form det(t1X1 + · · · + tmd2Xmd2) in R(nd,md2) aremapped to semi-invariants of the form fT for some T .

These steps allow us to reduce the theorem to the case d = 1. In the third step weprove the theorem for d = 1.

Step 1: Let U = Matdnd,n×Matmn,n×Matdn,nd and consider the map

ϕ : U = Matdnd,n×Matmn,n×Matdn,nd → Matmd2

nd,nd

(Y1, . . . , Yd, X1, . . . , Xm, Z1, . . . , Zd) )→ (YiXjZk | i, k ∈ [d], j ∈ [m]).

Later in the proof we will give an (non-linear) embedding ψ : Matmn,n → U and we

will show that ψ( ϕ( maps the semi-invariants of weight (−1, 1) in R(nd,md2) to thesemi-invariants of weight (−d, d) in R(n,m). First, we analyze the map ϕ.

U admits aGLnd×GLn×GLn×GLnd module structure. TheGLn×GLn action is givenas follows: setting V = W = Kn, we can identify Matn,n with Hom(V,W ) = V ∗ ⊗W .



Moreover we identify Matnd,n = W ∗ ⊗Knd and Matn,nd = V ⊗Knd. Then U becomesan GL(W )×GL(V )-module and in coordinates it is given by

(g, h) · (Y1, . . . , Yd, X1, . . . , Xm, Z1, . . . , Zd) = (. . . , Yig−1, . . . , gXjh

−1, . . . , hZk, . . . ).

The GLnd ×GLnd action is given by

(G,H) · (. . . , Yi, . . . , Xj , . . . , Zk, . . . ) = (. . . , GYi, . . . , Xj , . . . , ZkH−1, . . . ).

Observe that ϕ is constant on GL(W )×GL(V ) orbits since

ϕ((g, h) · (Y1, . . . , Yd, X1, . . . , Xm, Z1, . . . Zd)) = ϕ(. . . , Yig−1, . . . , gXjh

−1, . . . , hZk, . . . )

= (Yig−1gXjh

−1hZk | i, k ∈ [d], j ∈ [m])

= (YiXjZk | i, k ∈ [d], j ∈ [m])

= ϕ(Y1, . . . , Yd, X1, . . . , Xm, Z1, . . . , Zd)

Thus, ϕ( maps K[Matmd2

nd,nd] to K[U ]GLn×GLn In fact, without coordinates this map is

/Kd ⊗Knd ⊗W ∗0×

/W ⊗Km ⊗ V ∗0×

/V ⊗ (Knd)∗ ⊗Kd

0→ (Knd)∗ ⊗Knd ⊗Kmd2

given by the contraction of V and W . Now the First Fundamental Theorem of InvariantTheory for GLn (see Theorem 2.2.23 and the following remark) implies that

ϕ : U → Im(ϕ)

is in fact the categorical quotient

ϕ : U → U//(GLn ×GLn)

soϕ( : K[Matmd2

nd,nd] → K[U ]GLn×GLn

is surjective.

ϕ behaves nicely under the action of GLnd×GLnd too: ϕ is GLnd×GLnd-equivariantsince

ϕ((G,H) · (. . . , Yi, . . . , Xj , . . . , Zk, . . . )) = ϕ(. . . , GYi, . . . , Xj , . . . , ZkH−1, . . . )

= (GYiXjZkH−1 | i, k ∈ [d], j ∈ [m])

= (G,H) · (YiXjZk | i, k ∈ [d], j ∈ [m]).

Thus, ϕ maps the semi-invariants of Matmd2

nd,nd of weight (−1, 1) onto the semi-invariantsof U (under the action of GLnd ×GLnd) of weight (−1, 1), i.e.

ϕ( : SI(Matmd2

nd,nd, GLnd×GLnd, (−1, 1)) ↠ SI(U,GLnd×GLn×GLn×GLnd, (−1, 0, 0, 1)).



Now, we want to show that there is a surjection from SI(U,GLnd × GLn × GLn ×GLnd, (−1, 0, 0, 1)) to SI(Matmn,n, GLn ×GLn, (−d, d)).

Let (A1, . . . , Ad) ∈ Matdnd,n and (B1, . . . , Bd) ∈ Matdn,nd be such that

1A1 . . . Ad

2= Ind,

%

&'B1...

Bd

(

)* = Ind

and consider the map

ψ : Matmn,n → U

(X1, . . . , Xm) )→ (A1, . . . , Ad, X1, . . . , Xm, B1, . . . , Bd).

As a GLnd×GLnd-module, U is isomorphic to (Knd)∗⊕Matmn,n⊕(Knd) where the first

factor GLnd acts on the first summand (Knd)∗, the second factor GLnd acts on the thirdsummand Knd and Matmn,n is fixed by both groups. By the First Fundamental Theoremof Invariant Theory for SLn (see Theorem 2.2.24), we have

SI(U,GLnd ×GLnd, (−1, 1)) = detY FdetZ | F ∈ K[Matmn,n]

where

detY : U → K

(Y1, . . . , Yd, X1, . . . , Xm, Z1, . . . , Zd) )→ det1Y1 . . . Yd

2

and similarly detZ maps the same vector to

det

%

&'Z1...

Zd

(

)*

If F is a semi-invariant of weight (−d, d), then detY FdetZ is fixed under the action ofGLn ×GLn:

(detY FdetZ)(. . . , Yig−1, . . . , gXjh

−1, . . . , hZk, . . . )

= detY (Y1g−1, . . . , Ydg

−1) · F (gX1h−1, . . . , gXmh−1) · detZ(hZ1, . . . , hZd)

= det(g)−d det(h)ddetY (Y1, . . . , Yd)detZ(Z1, . . . , Zd)F (gX1h−1, . . . , gXmh−1)

= (detY FdetZ)(. . . , Yi, . . . , Xj , . . . , Zk, . . . ).

Thus, for a semi-invariant F of weight (−d, d)

detY FdetZ ∈ SI(U,GLnd ×GLn ×GLn ×GLnd, (−1, 0, 0, 1))



and by the choice of ψ, we have ψ((detY FdetZ) = F . Hence, ψ( gives a surjection

SI(U,GLnd ×GLn ×GLn ×GLnd, (−1, 0, 0, 1)) → SI(Matmn,n, GLn ×GLn, (−d, d)).

This is what we wanted to show. By composing ψ(ϕ(, we obtain a surjection from semi-invariants of Matmd2

nd,nd of weight (−1, 1) to semi-invariants of Matmn,n of weight (−d, d).

Step 2: ψ( ϕ( maps polynomials of the form det(t1X1+ · · ·+ tmXm) to polynomialsof the form fT :

Recall the definition of A1, . . . , Ad and B1, . . . , Bd from the previous step. A directcomputation shows that

AiXjBk = Eik ⊗Xj

where Eik denotes the matrix with every entry equals 0 except the i, k-th entry whichequals to 1.

Hence we have

ϕ ψ(X1, . . . , Xm) = (Eik ⊗Xj | i, k ∈ [d], j ∈ [m])

which implies that

ψ( ϕ((det(+

i∈[d]

+

j∈[m]

+

k∈[d]tijkXijk)) = det

# +

j∈[m]

(+

i,k∈[d]tijkEik)⊗Xj

$.

Set Tj =,

i,k∈[d] tijkEik. Then the latter polynomial equals det(T1 ⊗X1 + · · ·+ Tm ⊗Xm). The matrix T1⊗X1+ · · ·+Tm⊗Xm is permutation equivalent to X1⊗T1+ · · ·+Xm ⊗ Tm, i.e. there exists a permutation matrix P such that the conjugating the firstmatrix with P results in the second matrix. Thus, the given polynomial equals to fT upto a sign. Therefore, the image of det(t1X1 + · · ·+ tmXm) is of the form fT for some T ,which is what we wanted to show.

Step 3: Now we want to show that the invariants of weight (−1, 1) is spanned by theinvariants of the form det(t1X1 + · · ·+ tmXm).

Let V = W = Kn and identify Matmn,n with Hom(V,W )m = V ∗ ⊗W ⊗Km. First, weassume that char(K) = 0. Recall the first Cauchy formula (Theorem 2.1.24): There is aGL(A)×GL(B)-module isomorphism

Sk(A⊗B) ="

λ⊢kLλ(A)⊗ Lλ(B)

where Lλ(A) is the Schur functor corresponding to the partition λ. Applying the formulawe have

K[Matmn,n] ="

k

Sk(V ⊗ (W ⊗Km)∗) ="

λ

Lλ(V )⊗ Lλ((W ⊗Km)∗)



as a GL(W )×GL(V )-module.

We first compute the the subspace of semi-invariants of GL(V ) of weight det. EachLλ(V ) is an irreducible GL(V )-module, thus Lλ(V ) is a weight space of GL(V ) corre-sponding to the character det : GL(V ) → K× if and only if λ = (1n), i.e. Lλ(V ) =Λn(V ). We obtain

SI(Matmn,n, GL(V ), det) ⊆ L(1n)(V )⊗ L(1n)((W ⊗Km)∗)

= Λn(V )⊗ Λn((W ⊗Km)∗)

where SI(Matmn,n, GL(V ), det) denotes theGL(V )-semi-invariants ofK[Matmn,n] of weightdet : GL(V ) → K×.

Now we can further apply the second Cauchy formula

Λk(A⊗B) ="

λ⊢kLλ(A)⊗ Lλ′(B)

to obtain

SI(Matmn,n, GL(V ), det) ⊆ Λn(V )⊗ Λn((W ⊗Km)∗)

="

λ⊢nΛn(V )⊗ Lλ(W

∗)⊗ Lλ′((Km)∗).

Again, Lλ(W∗) has weight det−1 if and only if λ = (1n) so we have

SI(Matmn,n, GL(W )×GL(V ), (det−1, det) ⊆ Λn(V )⊗ Λn(W ∗)⊗ Sn((Km)∗)

However, it is clear that GL(W )×GL(V ) acts on the right-hand side with weight (−1, 1)so we have the equality.

Let v1, . . . , vn be the (standard) basis of V = Kn and ϕ1, . . . ,ϕn be the (standard)basis of W ∗ = (Kn)∗. Then Λn(V ) is spanned by v1 ∧ · · · ∧ vn and Λn(W ∗) is spannedby ϕ1 ∧ · · · ∧ ϕn.

Now consider the tensor

v1 ∧ · · · ∧ vn ⊗ ϕ1 ∧ · · · ∧ ϕn ⊗ (e1)⊗n ∈ Λn(V )⊗ Λn(W ∗)⊗ Sn((Km)∗)

where e1 is the first vector of the dual basis of the standard basis of Km. Then

(v1 ∧ · · · ∧ vn ⊗ ϕ1 ∧ · · · ∧ ϕn ⊗ (e1)⊗n)(X1, . . . , Xm) =!

σ∈Sn

sgn(σ)ϕσ(i)(X1vi)

=!

σ∈Sn

sgn(σ)(X1)iσ(i)

= det(X1).



To finish the proof, we introduce a new action on Matmn,n: GLm acts on V ∗⊗W ⊗Km

by acting on the third factor Km. The actions of GL(W )×GL(V ) and GLm commute.Now Sn((Km)∗) is an irreducible GLm-module so if e1 is the first vector of the standarddual basis of (Km)∗, then we have

Sn((Km)∗) = 〈GLm · (e1)⊗n〉

As a result, we have

Λn(V )⊗ Λn(W ∗)⊗ Sn((Km)∗) = 〈v1 ∧ · · · ∧ vn ⊗ ϕ1 ∧ · · · ∧ ϕn ⊗ (GLm · (e1)⊗n)〉

which implies that

SI(Matmn,n, GL(W )×GL(V ), (det−1, det)) = 〈GLm · det(X1)〉= 〈det(t1X1 + · · ·+ tmXm) | t1, . . . , tm ∈ K×〉.

This finishes the proof in characteristic 0. For char(K) > 0, we use the Cauchyformulas in arbitrary characteristic (see Theorem 2.1.30 and 2.1.36). Using the formulasas in the previous case, we again have

SI(Matmn,n, GLn ×GLn, (det−1, det)) ∼= Λn(V )⊗ Λn(W ∗)⊗Dn((Km)∗)

where Dn is the divided power functor. Introducing the GLm action again we obtainthe same result : Λn(V )⊗Λn(W ∗)⊗Dn((Km)∗) pulls back to the subspace spanned bythe polynomials of the form

det(t1X1 + · · ·+ tmXm), ti ∈ K×.

3.3 Degree Bounds for Generators of S(n,m) and R(n,m)

In this section, we will give degree bounds for a generating set of invariants of S(n,m).We fix the following notation:

Definition 3.3.1. Given a reductive group G and a rational G-module V , we defineβ(K[V ]G) to be the smallest integer D such that the invariants of degree ≤ D generatethe invariant ring, i.e.

β(K[V ]G) = minD | ∪Dd=1K[V ]Gd generates K[V ]G

Definition 3.3.2. A set S ⊆ K[V ]G of invariants is said to be a separating set ofinvariants if G · v ∩ G · w = ∅ implies the existence of an invariant f ∈ S such thatf(v) ∕= f(w).



We define βsep(K[V ]G) to be the smallest integer D such that the invariants of degree≤ D form a separating set of invariants.

Recall that we defined the nullcone NG(V ) of the action of G on V as the zero set ofall non-constant homogeneous invariants.

Definition 3.3.3. We define γ(K[V ]G) to be the smallest integer D such that the zeroset of non-constant invariants of degree D defines NG(V ), i.e.

minD | NG(V ) = V (∪Dd=1K[V ]Gd )

Lemma 3.3.4. For a reductive group G and a rational G-module V we have

γ(K[V ]G) ≤ βsep(K[V ]G) ≤ β(K[V ]G)

Proof. Observe that a set of generators for the invariant ring forms a separating set andthe zero set of any separating set is necessarily equals the nullcone.

We prove in the previous section that S(n,m) is generated by Trw for w ∈ [m]∗. In[Raz74], Razmyslov studied trace identities and as a result he obtained the followingtheorem in characteristic 0.

Theorem 3.3.5 ([Raz74]). Assume that char(K) = 0. Then β(S(n,m)) ≤ n2.

We unfortunately skip the proof of this theorem as it involves some techniques we didnot introduce. We continue with a bound for γ(R(n,m)) by Derksen and Makam:

Theorem 3.3.6 ([DM15] ). Let n ≥ 2 and (X1, . . . , Xm) ∈ Matmn,n and further assumethat (X1, . . . , Xm) is not in the nullcone of the left-right action. Then for all d ≥ n− 1,there exists T = (T1, . . . , Tm) ∈ Matd,d such that

fT (X1, . . . , Xm) = det(X1 ⊗ T1 + · · ·+Xm ⊗ Tm) ∕= 0

Proof. Set X = 〈X1, . . . , Xm〉. Since (X1, . . . , Xm) is not in the nullcone, there existsd such that rk(X ⊗ Matd,d) = dn. Without loss of generality we may assume thatd ≥ n − 1. Then applying Lemma 3.2.22 repeatedly, we obtain rk(X ⊗Matn−1,n−1) =n(n − 1) which implies that there exists T = (T1, . . . , Tm) ∈ Matmn−1,n−1 such thatfT (X1, . . . , Xm) ∕= 0.

Corollary 3.3.7. For n ≥ 2 we have γ(R(n,m)) ≤ n(n− 1).

Proof. Assume that (X1, . . . , Xm) ∈ Matmn,n is not in the nullcone of the left-right ac-tion. By the previous theorem, there exists T = (T1, . . . , Tm) ∈ Matmn−1,n−1 such thatfT (X1, . . . , Xm) ∕= 0. Thus, for all X = (X1, . . . , Xm) ∈ Matmn,n that is not in the null-cone, there exists a matrix semi-invariant f of degree n(n − 1) such that f(X) ∕= 0. Inother words, if X is in the zero set of all matrix semi-invariants of degree n(n− 1), thenX must be in the nullcone. Hence, γ(R(n,m)) ≤ n(n− 1).



The invariant ring K[V ]G of a rational module V and a linearly reductive group G isCohen-Macaulay by Hochster-Roberts Theorem (see, for example, [Stu93] Section 2.3 or[DK02] Section 2.6.2). In [DK02], the following is proven:

Proposition 3.3.8 ([DK02], Corollary 2.7.3). Let G be a linearly reductive group andV be a representation. Let f1, . . . , fr be a homogeneous system of parameters and letdi = deg(fi). Then

β(K[V ]G) ≤ maxd1 + d2 + · · ·+ dr − r, d1, d2, . . . , dr.

In characteristic 0 we obtain the following:

Theorem 3.3.9 ([DM15]). Assume that char(K) = 0. Then β(R(n,m)) ≤ mn4.

Proof. First assume that n ≥ 2. By Corollary 3.3.7, the invariants of degree ≤ n2 − ndefine the nullcone. Let r be the Krull dimension of R(n,m). Note that since Matmn,nhas dimension mn2, we have r ≤ mn2. Using Noether Normalization Lemma (see, forexample, [Eis95], Theorem A1) there exist invariants f1, . . . , fr of degree n2 − n thatform a homogeneous system of parameters. If r = 1, then the above theorem impliesthat β(R(n,m)) ≤ n2 − n ≤ mn4. If r > 1 then

β(R(n,m)) ≤ r(n2 − n)− r = r(n2 − n− 1) ≤ mn2(n2 − n− 1) < mn4

For n = 1, R(1,m) is spanned in degree 1 so again β(R(n,m)) ≤ mn4.

Remark 3.3.10. When char(K) > 0, SLn × SLn is not linearly reductive so Hochster-Roberts theorem does not apply.

For a rational module V of dimension n over algebraically closed fields of characteristic0, Weyl’s polarization theorem (see [Wey66], Theorem 2.5A) states that the ring ofinvariants K[V ⊕m]G,m ≥ n is spanned by polynomials of the form

f(λ11v1 + · · ·+ λ1

mvm, . . . ,λn1v1 + · · ·+ λn

mvm)

where f ∈ K[V ⊕n]G.

Example 3.3.11. Consider the action of SLn on V = Kn. We can identify V ⊕m with thematrix space Kn×m where SLn acts via left multiplication. By the First FundamentalTheorem of Invariant Theory for SLn, for m < n, the invariant ring K[V ⊕m]SLn containsonly constants. For m = n, however, there is a non-constant invariant:

K[V ⊕n]SLn = K[Matn,n]SLn = K[det].

Now, Weyl’s polarization theorem states that for any m > n, the ring of invariantsK[V ⊕m]SLn is spanned by the elements of the form

X =1v1 v2 . . . vm

2)→ det

1,mi=1 λ

1i vi . . .

,mi=1 λ

ni vi

2= det(X · g)



for some g ∈ Matm,n. Note that each n× n-minor of X can be obtained this way whenwe take

g =

7In0

8.

In fact, as the First Fundamental Theorem of Invariant Theory for SLn implies, theinvariant ring K[V ⊕m]SLn is spanned by n× n-minors of X.

Remark 3.3.12. Weyl’s polarization theorem has the immediate consequence that form ≥ n = dimV we have

β(K[V ⊕m]G) ≤ β(K[V ⊕n])G.

We can use Weyl’s polarization theorem to replace the bound mn4 with n6:

Corollary 3.3.13. Assume that char(K) = 0. Then β(R(n,m)) ≤ n6.

Proof. Note that dimMatn,n = n2 so for m ≥ n2 Weyl’s polarization theorem im-plies that R(n,m) is spanned by the polarizations of invariants in R(n, n2), henceβ(R(n,m)) ≤ β(R(n, n2)) ≤ n6.

For char(K) ∕= 0, Derksen and Makam ([DM16b]) prove the above results by using ageneralization of Proposition 3.3.8. Hence we have

Theorem 3.3.14 ([DM16b], [DM18b]). The invariant ring R(n,m) is generated indegree ≤ mn4. Moreover, if char(K) > 2n6 + n2 then β(R(n,m)) ≤ n6.

Remark 3.3.15. In [DM15], Derksen and Makam prove the lower bound n⌊√n+ 1⌋ ≤

γ(R(n,m)). In a way, this lower bound implies that there is not much space to strengthenthe given upper bound. However, for β(R(n,m)) known lower bounds are much smallerthan the upper bound mn4.

There is a useful surjection R(n,m+ 1) → S(n,m) considered by Domokos:

Lemma 3.3.16. Consider the map

ϕ : Matmn,n → Matm+1n,n

(X1, . . . , Xm) )→ (I,X1, . . . , Xm)

Then ϕ( maps R(n,m+ 1) onto S(n,m).

Proof. Given a matrix semi-invariant f ∈ R(n,m+1), set F = ϕ(f . Then F is a matrixinvariant since

(g · F )(X1, . . . , Xm) = F (g−1X1g, . . . , g−1Xmg)

= f(I, g−1X1g, . . . , g−1Xmg)

= ((g, g) · f)(I,X1, . . . , Xm)

= f(I,X1, . . . , Xm)

= F (X1, . . . , Xm)



Here (g, g) · f = f since every semi-invariant has weight (−d, d) for some d ∈ Z.

Conversely, given a matrix invariant F ∈ S(n,m), consider

f : Matm+1n,n → K

(X1, . . . , Xm+1) )→ F (Adj(X1)X2, . . . ,Adj(X1)Xm+1)

Then (ϕ(f)(X1, . . . , Xm) = f(I,X1, . . . , Xm) = F (X1, . . . , Xm) hence ϕ(f = F . More-over, f is a matrix semi-invariant as

((g, h) · f)(X1, . . . , Xm+1) = f(g−1X1h, . . . , g−1Xm+1h)

= F (Adj(g−1X1h)g−1X2h, . . . ,Adj(g−1X1h)g

−1Xm+1h)

= F (h−1Adj(X1)gg−1X2h, . . . , h

−1Adj(X1)gg−1Xm+1h)

= F (h−1Adj(X1)X2h, . . . , h−1Adj(X1)Xm+1h)

= (h · F )(Adj(X1)X2, . . . ,Adj(X1)Xm+1)

= F (Adj(X1)X2, . . . ,Adj(X1)Xm+1)

= f(X1, . . . , Xm+1)

and the result follows.

Theorem 3.3.17 ([DM16b]). We have β(S(n,m)) ≤ (m+ 1)n4.

Proof. By the description of ϕ in the above lemma, it is easy to see that ϕ( is a degreepreserving map. Thus β(S(n,m)) = β(R(n,m+ 1)) ≤ (m+ 1)n4.

We collect the results in one theorem:

Theorem 3.3.18. 1. If char(K) = 0, then β(S(n,m)) ≤ n2.

2. β(S(n,m)) ≤ (m+ 1)n4 in arbitrary characteristic.

3. β(R(n,m)) ≤ mn4 in arbitrary characteristic. Moreover if char(K) = 0 orchar(K) > 2n6 + n2 then β(R(n,m)) ≤ n6.

4. γ(R(n,m)) ≤ n(n− 1) in arbitrary characteristic.

3.4 Algorithms

In this section, our aim is to give polynomial time algorithms for the orbit closure problemfor simultaneous conjugation and the left-right action. To this end, we will start by thealgorithm for the computation of non-commutative rank of matrix spaces by Ivanyos,Qiao and Subrahmanyam ([IQS15b], [IQS15a]) and use this algorithm (which we willcall IQS algorithm) to solve orbit closure problems for simultaneous conjugation andleft-right action. The algorithms for the orbit closure intersection problems are given byDerksen and Makam in [DM18a].



3.4.1 The Constructive Regularity Lemma

In this section, we present a constructive proof of the so-called regularity lemma whichwas already mentioned in Section 3.3. This lemma will be the heart of the algorithmthat we present in the next section.

Lemma 3.4.1 (Regularity Lemma). Let X ⊆ Matn,n be a subspace and d ∈ Z≥1. Thenthe rank of the space X d = X ⊗Matd,d is divisible by d.

The lemma is due to Ivanyos, Qiao and Subrahmanyam. In [IQS15b], the authorsgive a proof of lemma for fields of large enough cardinality. However, as we always workwith algebraically closed fields, we do not need this assumption. Moreover, in the samepaper, authors give a constructive version of the lemma for fields of characteristic notdivisible by d. Later, in [IQS15a], they remove this condition. We give the followingversion of the lemma which we will call The Constructive Regularity Lemma:

Lemma 3.4.2 ([IQS15b] Lemma 5.7, [IQS15a] Lemma 4.1). Let X ⊆ Matn,n be asubspace and assume that we have a given matrix A ∈ X d of rank rk(A) > (r − 1)dfor some r ≤ n. Then there exists a deterministic algorithm that returns a matrixA′ ∈ X d of rank rk(A′) ≥ rd. This algorithm uses poly(nd) arithmetic operationsand if X is defined over Q, the algorithm runs in polynomial time. In particular, everyintermediate number of the algorithm have bit length polynomial in the input size.

Remark 3.4.3. Observe that Lemma 3.4.2 implies Lemma 3.4.1: Let A ∈ X d be amatrix of maximal rank, i.e. rk(A) = rk(X d). If rk(A) is not divisible by d, then thereexists a, b such that rk(A) = ad+ b and b < d. Now apply Lemma 3.4.2 with a = r − 1.The algorithm returns a matrix A′ ∈ X d of rank at least (a + 1)d ≥ rk(A). This is acontradiction as A has maximal rank.

The algorithm (somewhat surprisingly) depends on effective constructions of centraldivision algebras. We start with some definitions:

Definition 3.4.4. A division algebra D over K is an (not necessarily commutative) as-sociative K-algebra in which every non-zero element has a multiplicative inverse. Givena division algebra D, the opposite division algebra Dop is defined to be the algebra withthe same set of elements as D but the multiplication x · y of Dop is defined to be y ∗ xwhere ∗ is the multiplication of D.

The center of D is defined as

Z(D) = x ∈ D | xy = yx for all y ∈ D

and it is a field extension of K. Throughout the section, we will always assume that Dis finite dimensional as a Z(D) vector space. If Z(D) = K, then D is called a centraldivision algebra over K. In this case, the dimension of D as a K-vector space is d2 forsome d which is called the index of D.



The importance of central division algebras for the regularity lemma is given by thefollowing surprising property of central division algebras:

Lemma 3.4.5 ([IQS15b], Claim 5.5). Let K be a field and let Γ be a K(Y )-vector spacebasis of Matd,d(K(Y )) such that K(Y d)-linear span D of Γ is a central division algebraof index d over K(Y d). Then, every matrix in

Matn,n(K)⊗K D ⊆ Matnd,nd(K(Y ))

has rank divisible by d.

We will prove this fact inside the proof of the main theorem in the end of this section.We first expand our knowledge on division algebras.

Proposition 3.4.6. If D is a central division algebra of index d and x, y ∈ D, thenthere is a linear transformation

µx,y : D → D

z )→ x ∗ z ∗ y.

The map

ψ : D ⊗Dop → Matd2,d2

sending x ⊗ y to µx,y is an isomorphism of division algebras. Moreover, the image ofD ⊗ 1 and 1⊗Dop commute in Matd2,d2.

Proof. For the first part, see [Lam91], Corollary 15.5. The second part follows easily aswe have

µx,1(µ1,y(v)) = x ∗ z ∗ y = µ1,y(µx,1(v)).

Definition 3.4.7. Given a field K, a cyclic extension L of K is a finite Galois extensionof K having a cyclic Galois group.

The algorithm that we will introduce in this section requires the effective constructionof cyclic extensions of the function field K(X). By the construction of a cyclic extension,we mean giving a K-vector space basis A1, . . . , As of L together with the structureconstants. Recall that the structure constants γkij ∈ K are given by the equations

AiAj =

s+

k=1

Ak.

In [IQS15b], efficient construction of a cyclic extension of K(X) of degree d is givenunder the condition that char(K) does not divide d. Later, in [IQS15a], the authorsremove this condition.



Lemma 3.4.8 ([IQS15b], [IQS15a]). Let K be a field. Let d be any non-negative integer.If char(K) = 0, set d1 = d. If char(K) = p > 0, let d1 be the p-free part of d, that is,d = d1p

s for some s where p ∤ d1. Assume that K contains a known primitive d1-throot of unity ξ and let X be a formal variable. Then a cyclic extension of L havingdegree d over K(X) can be computed using poly(d) many arithmetic operations. L willbe given by structure constants with respect to a basis and the matrix for a generator ofthe cyclic Galois group L/K(X) in terms of the same basis will also be given. All theoutput entries (the structure constants as well as the entries of the matrix representingthe Galois group generator) will be polynomials of degree poly(d) in K[X]. Furthermore,for K = Q[ d1

√1], the bit complexity of the algorithm (as well as the size of the output)

is poly(d).

Proof. Let L1 = K(Y ) and set X = Y d1 . Then it is clear that 1, Y, Y 2, . . . , Y d1−1 forma K(X) basis for L1 with Y iY j = Y i+j if i + j < d1 and Y iY j = XY i+j−d1 otherwise.Moreover, the matrix

σ =

%

&&&'

1ξ

. . .

ξd1−1

(

)))*

is a K(X)-automorphism of L1 of degree d1, mapping Y j to ξjY j .

Observe that if char(K) = 0 or does not divide d, then d = d1 and we can take L = L1.We are done in this case.

Now, assume that char(K) = p > 0 divides d. Let Fp be the prime field of K. Assumethat we have constructed a cyclic extension L2 of Fp(X) of degree ps together with amatrix σ2 that generates the cyclic Galois group of the extension L/Fp(X). Set

L = L1 ⊗Fp(X) L2.

Observe that L contains K(X) as K(X) ∼= K(X) ⊗Fp(X) Fp(X). Note that L hasdimension d as a K(X)-vector space as [L1 : K(X)] = d1 and [L2 : Fp(X)] = ps.Moreover, the matrix σ1 ⊗ σ2 has order d1p

s = d.

Note that we still need the construction of a cyclic extension of Fp(X) of degree ps.We give the following lemma without proof:

Lemma 3.4.9 ([IQS15a], Lemma 3.1). Given a prime p and an integer s ≥ 1, one canconstruct in time poly(ps) a cyclic extension Ks of Fp(X) of degree ps such that Fp isintegrally closed in Ks. The field Ks will be given in terms of structure constants withrespect to a basis over Fp(X), and the generator σ for the Galois group will be given byits matrix in terms of the same basis. The structure constants as well as the entries ofthe matrix for σ will be polynomials in Fp[X] of degree poly(ps).

The following proposition connects central division algebras with cyclic field exten-sions:



Proposition 3.4.10 ([IQS15b], Fact 4.3). Let K be a field and let L be a cyclic extensionof K of degree d. Let σ be a generator of the Galois group, and Z be transcendental overL. Then σ extends to an automorphism (denoted again by σ) of L(Z) such that thefixed field of σ is K(Z). Thus, L(Z) is a cyclic extension of K(Z). Let D be the K(Z)-algebra generated by a basis of L over K and by an element U with relations Ud = Zand Ua = σ(a)U for every a ∈ L(Z). Then D is a central division algebra of index dover K(Z).

Moreover, this proposition can be constructivized:

Proposition 3.4.11 ([IQS15b], Proposition 4.4). Let L be a cyclic extension of degreed of a field K and suppose that L is given by structure constants with respect to a K-basis A1, . . . , Ad. Similarly, assume that a generator σ of the Galois group is given byits matrix with respect to the basis A1, . . . , Ad. Let Y be a formal variable. Then usingpoly(d) arithmetic operations in K, one can construct a K(Y )-basis Γ of Matd,d(K(Y ))such that the K(Y d)-linear span of Γ is a central division algebra over K(Y d) of indexd.

Proof. Set Z = Y d and let D be a central division algebra over K(Z) as in the aboveproposition. We can construct a D′ ⊆ Matd,d(K(Y )) isomorphic to D as follows: Theelements AiU

j , i, j = 1, . . . , d form a K(Z)-basis of D.

Now we can give the main theorem of this section, that is the regularity lemma:

Theorem 3.4.12 ([IQS15b], Lemma 5.4). Let (X1, . . . , Xm) ∈ Matmn,n(K) and set X =

〈X1, . . . , Xm〉. Assume that we are given a matrix A ∈ X d in some blow-up of X suchthat rk(A) = (r− 1)d+ k for some 1 < k < d. Let X and Y be formal variables and setL = K(X) and assume that we are given a K(X,Y )-basis Γ of Matd,d(K(X,Y )) suchthat the K(X,Y d)-linear span of Γ is a central division algebra D over K(X,Y d). Letδ be the maximum of degrees of polynomials appearing as numerators or denominatorsof the entries of the matrices in Γ. Then using poly(nd+ δ) arithmetic operations in K,one can find a matrix A′′ ∈ X d with rk(A′′) ≥ rd. Furthermore, if (X1, . . . , Xm) aredefined over Q, the bit complexity of the algorithm is polynomial in the size of the inputdata.

Proof. Let X d = X⊗Matd,d(K) as before and set X d(K(X,Y )) = X⊗KMatd,d(K(X,Y )).Note that K(X,Y )-linear span of X d is X d(K(X,Y )) as we have

〈X d〉K(X,Y ) = X d ⊗K K(X,Y ) = X ⊗K Matd,d(K)⊗K K(X,Y ) = X d(K(X,Y )).

Moreover, the K(X,Y )-linear span of X ⊗D is also X d(K(X,Y )) since D contains abasis of Matd,d(K(X,Y )).

Claim : Every matrix in Matn,n(K) ⊗K D ⊆ Matnd,nd(K(X,Y )) has rank divisibleby d.



Proof : Since D is a K(X,Y d)-algebra, we have the equality Matn,n(K) ⊗K D =Matn,n(K(X,Y d)) ⊗K(X,Y d) D. On the other hand, D ⊆ Matd,d(K(X,Y )) so it acts

on the space K(X,Y )d ∼= K(X,Y d)d2. Hence, we have a natural action of the space

Matn,n(K(X,Y d))⊗K(X,Y d)D on K(X,Y d)n⊗K(X,Y )d ∼= K(X,Y d)n⊗K(X,Y d)d2 ∼=

K(X,Y d)nd2. Now, Dop commutes with this action by Proposition 3.4.6. Thus, for any

x ∈ Dop and A′ ∈ Matn,n(K)⊗K D we get

x(A′ ·K(X,Y )dn) = A′ · (xK(X,Y )dn) ⊆ A′ ·K(X,Y )dn

so the image A′ ·K(X,Y )dn of A′ is a Dop-module. Thus, the dimension of A′ ·K(X,Y )dn

overK(X,Y d) is divisible by d2 which implies that its dimension overK(X,Y ) is divisibleby d. Since this dimension equals to the rank of A′, the result follows.

We will use this claim for the algorithm. Since Γ = C1, . . . , Cd2 is a basis ofMatd,d(K(X,Y )), we can write

A =+

i∈[m],j∈[d2]

λijXi ⊗ Cj , λij ∈ K(X,Y ).

Now, A may or may not be in Matn,n(K)⊗K D as the coefficents λij ∈ K(X,Y ) are notnecessarily in K. However, this is not a problem as we can replace the coefficients λij

with µij ∈ K without decreasing the rank:

Claim : We can compute µij ∈ K such that

A′ =+

i∈[m],j∈[d2]

µijXi ⊗ Cj

satisfies rk(A′) ≥ rk(A).

Proof : Without loss of generality assume that λ11 ∕∈ K. Consider the matrix

B(x) = xX1 ⊗ C1 + λ12X1 ⊗ C2 + · · ·+ λmd2Xm ⊗ Cd2

with entries in K(X,Y )[x]. Let M be the rk(A)× rk(A)-submatrix of A with non-zerodeterminant. Then, the determinant of the same block in B is a non-zero polynomial inx of degree at most rk(A). By evaluating at rk(A) + 1 many points, we find µ11 ∈ Kthat is not a zero of this polynomial. Thus, the matrix B(µ11) has rank at least rk(A).We can similarly replace each λij ∈ K(X,Y ) with some coefficient µij ∈ K such thatthe corresponding matrix

A′ =+

i∈[m],j∈[d2]

µijXi ⊗ Cj

has rank at least rk(A).

Note that A′ ∈ Matn,n(K)⊗K D, so its rank is divisible by d. Since rk(A) > (r − 1)dand rk(A′) ≥ rk(A), we deduce that rk(A′) ≥ rd.



Now, we want to write A′ as a matrix in X d over K. To this end, we expand A′ as

A′ =+

i∈[m],j,k∈[d]νijkXi ⊗ Ejk, νijk ∈ K(X,Y )

where Ejk denotes the basis elements of Matd,d(K(X,Y )). Then, we apply the sameprocess above to replace each coefficient νijk with a coefficient ρijk ∈ K such that thematrix

A′′ =+

i∈[m],j,k∈[d]ρijkXi ⊗ Ejk

satisfies rk(A′′) ≥ rk(A′) ≥ rd. A′′ is the matrix we wanted to compute.

input : X1, . . . , Xm ∈ Matn,n, a matrix A ∈ 〈X1, . . . , Xm〉d of rank> (r − 1)d for some r

output : A matrix A′′ ∈ 〈X1, . . . , Xm〉d of rank ≥ rd.initialize: ξ ← a primitive d-th root of unity in K

1 X,Y ← two formal variables2 Γ ← C1, . . . , Cd2, a K(X,Y )-basis of Matd,d(K(X,Y )) such that

K(X,Y d)-linear span of Γ is a central division algebra of index d over K(X,Y d)(see Proposition 3.4.11)

3 Write A =,

i∈[m],j∈[d2] λijXi ⊗ Cj , λij ∈ K(X,Y )

4 A′ ←,

i∈[m],j∈[d2] µijXi ⊗ Cj , µij ∈ K such that rk(A′) ≥ rd

5 Write A′ =,

i∈[m],j,k∈[d] νijkXi ⊗ Ejk, νijk ∈ K(X,Y )

6 A′′ ←,

i∈[m],j,k∈[d] ρijkXi ⊗ Ejk, ρijk ∈ K such that rk(A′′) ≥ rk(A′)

7 return A′′

Algorithm 1: The algorithm for the constructive regularity lemma

Remark 3.4.13. There are two issues we did not consider in the algorithm:

1. Observe that the second line of the algorithm requires a known primitive d-th rootof unity ξ. Since we always assume that K is an algebraically closed field, theexistence of ξ is clear. However, algorithmically, the computation of ξ may nothave a polynomial time algorithm. In [IQS15b], Section 4.4.1, the authors describea procedure to overcome this issue.

2. In the algorithm, one computes the rank of matrices over the function fieldK(X,Y ).There is in fact an algorithm ([IQS15b], Proposition 4.8) that computes the rankof such matrices using poly(N,D) many arithmetic operations over K where N isthe size of the matrices and D is the maximal degree appearing in numerators anddenominators of the entries of the matrix.



3.4.2 The Computation of Non-Commutative Rank

In this section, we show that there exists a polynomial time algorithm for the computa-tion of the so-called non-commutative rank. The heart of the algorithm is the construc-tive regularity lemma from the previous section. We start with a definition.

Definition 3.4.14. Let X ⊆ Matn,n be a subspace. The non-commutative rank ncrk(X )of X is defined as

ncrk(X ) = n−maxc | there exists a c-shrunk subspace of X.

Here, we call a tuple (U, V ) of subspaces of Kn a c-shrunk subspace of X if X (U) ⊆ Vand dimU − dimV = c.

Observe that if X1, . . . , Xm is a spanning set of X , then ncrk(X ) < n if and onlyif the tuple (X1, . . . , Xm) is in the nullcone of the left-right action. This follows by thefollowing equivalence from previous sections:

Theorem 3.4.15. Let (X1, . . . , Xm) ∈ Matmn,n and X = 〈X1, . . . , Xm〉 ⊆ Matn,n. Thenthe following are equivalent:

1. (X1, . . . , Xm) is in the nullcone of the left-right action.

2. There exist subspaces U, V ⊆ Kn such that

∀i,Xi(U) ⊆ V and dimV < dimU,

i.e. (X1, . . . , Xm) is a compression space.

3. For all d ∈ Z≥1 we have

rk(X d) < dn

where X d = X ⊗ Matd,d is the d-th tensor blow-up of X and rk(X d) denotesthe maximum rank among matrices in X d.

In Section 3.2.1, we discussed a concept called Wong sequences that helps us to detectthe existence of subspaces of the form given in the second item of the above theorem.Recall that given a subspace X ⊆ Matn,n and a matrix A ∈ X , the second Wong sequenceof (A,X ) is defined as

W0 = 0, W1 = X (A−1(W0)), . . . , Wi+1 = X (A−1(Wi)), . . .

We proved that this sequence stabilizes: There exists an index l such that W0 ⊆ W1 ⊆. . .Wl−1 ⊆ Wl = Wl+1 = Wl+2 = . . . We call Wl the limit of the sequence and denote itby W ∗. Moreover, recall Lemma 3.2.13: There exist a n− rk(A)-shrunk subspace if andonly if W ∗ ⊆ Im(A). We want to check this condition efficiently.

Remark 3.4.16. One can directly compute the second Wong sequence of (A,X ) just bynaive linear algebra. However, over Q for example, this may not result in a polynomial



time algorithm: The length of the second Wong sequence is bounded by n, and thecomputation of Wi+1 = X (A−1(Wi)) may lead to an explosion in the bit-size, i.e. thebit-sizes of entries of the vectors in Wi+1 can grow exponentially with respect to n.Therefore, we need a better way to compute the second Wong sequence.

Definition 3.4.17. We call a matrix B ∈ Matn,n a pseudo-inverse of the matrix A ifthe restriction of B to Im(A) is the inverse of the restriction of A to a direct complementof ker(A).

Remark 3.4.18. Observe that given A ∈ Matn,n, a pseudo-inverse B of A can be com-puted efficiently: We pick direct complements U and V to ker(A) and Im(A), respec-tively. Then we pick a map B0 : U → Im(A) such that BA =idIm(A) and pick anarbitrary map B1 : ker(A) → Kn. Then B = B0 ⊕B1 is a pseudo-inverse of A.

We state the following lemma without proof. The interested reader may check [IQKS13].

Lemma 3.4.19 ([IQKS13], Lemma 10). Let A ∈ X ⊆ Matn,n and let B be a pseudo-inverse of A. There exists a n− rk(A)-shrunk subspace of X if and only if

(XB)i(ker(AB)) ⊆ Im(A)

for all i. In the algebraic RAM model as well as over Q, this can be tested in deterministicpolynomial time. In this case, B(W ∗) and X (B(W ∗)) form a n−rk(A)-shrunk subspace.These spaces can also be constructed efficiently.

Unfortunately, the above lemma does not immediately give an algorithm for the Null-cone Membership Problem: Observe that if W ∗ ∕⊆ Im(A), then there do not exist an − rk(A)-shrunk subspace of X . However, there might be a c-shrunk subspace wherec < n − rk(A). To use the above lemma iteratively, we need to pick a matrix A′ ∈ Xof rank larger than rk(A) and check whether there exists a n− rk(A′)-shrunk subspace.On the other hand, the following example shows that it is not always possible to findsuch a matrix:

Example 3.4.20. Let X be the space spanned by the matrices

A =

%

&&'

0 1 0 0−1 0 0 00 0 0 00 0 0 0

(

))* , B =

%

&&'

0 0 1 00 0 0 0−1 0 0 00 0 0 0

(

))* , C =

%

&&'

0 0 0 00 0 1 00 −1 0 00 0 0 0

(

))* .

Direct computation shows that the second Wong sequence of (A,X ) is

W0 = 0, W1 = 〈(−1, 0, 0, 0), (0,−1, 0, 0)〉, W2 = W ∗ = K4.

Since W ∗ ∕⊆ Im(A), we deduce that X has no 2-shrunk subspace. On the other hand, ithas a 1-shrunk subspace: The vector (0, 0, 0, 1) is a common kernel of A,B and C. Wecannot certify this fact using Lemma 3.2.13 as there is no matrix A′ ∈ X of rank 3.



In the special case that X is spanned by two matrices, we can certify Nullcone Mem-bership directly using Wong sequences:

Lemma 3.4.21 ([IQKS13], Fact 11). Let X = 〈A,B〉 ⊆ Matn,n. Then rk(A) = rk(X )if and only if W ∗ ⊆ Im(A). Moreover, if W ∗ ∕⊆ Im(A), then we can efficiently findλ, µ ∈ K such that rk(λA+ µB) > rk(A).

The following theorem makes use of the above fact by reducing X to a 2-dimensionalsubspace. The construction of a matrix of rank larger than rk(A) is also given in theproof of the theorem.

Theorem 3.4.22 ([IQS15a], Theorem 5.1). Let X ⊆ Matn,n be a subspace. Assume thatwe are given a matrix A ∈ X d of rank rk(A) = rd. Then there exists a deterministicalgorithm that returns either an (n − r)d-shrunk subspace for X d (or equivalently an(n− r)-shrunk subspace for X ), or a matrix A′ ∈ X ⊗Matd′,d′ of rank at least (r+1)dd′

where d′ = r + 1. Furthermore, in the latter case an (r + 1) × (r + 1)-block matrix isalso found such that the corresponding (r + 1)dd′ × (r + 1)dd′ sub-matrix of A′ has fullrank. This algorithm uses poly(ndd′) arithmetic operations and over Q, all intermediatenumbers have bit lengths polynomial in the input size.

Proof. Step 1: We start by checking the existence of a (n − r)d-shrunk subspace. Tothis end, we compute the second Wong sequence

W0 ⊆ W1 ⊆ · · · ⊆ Wl = W ∗

of (A,X d) and their inverse images

Vi = A−1(Wi).

If W ∗ ⊆ Im(A), then U = A−1(W ∗) is an nd − rk(A) = (n − r)d-shrunk subspace ofX d. We note that the dimension of Wi increases by at least d at each step: As

(In ⊗Matd,d)X d = (In ⊗Matd,d)(X ⊗Matd,d) = X ⊗Matd,d = X d.

Thus, we have (I ⊗Matd,d)XWi = XWi. A subspace W satisfying (In⊗Matd,d)W = Wsatisfies W = W0 ⊗ Id for some subspace W0. Thus, dimWi is divisible by d. Hence,until stabilization, the dimension of dimWi increases by at least d. In particular, wehave l ≤ r + 1.

Observe that we can reduce U to a (n − r)-shrunk subspace of X : For each vectorv ∈ U ⊆ Knd write v as a d× n-matrix

v =

%

&&&&'

v1 vd+1 . . . v(n−1)d+1

v2. . .

......

. . ....

vd . . . . . . vnd

(

))))*.



Define U ′ to be the subspace of Kn spanned by the rows of v for each v ∈ U . Similarly,let V ′ be the subspace of Kn spanned by the rows of v for each v ∈ X d(U). Then(U, V ) is an (n− r)-shrunk subspace for X .

Step 2: Now assume that W ∗ ∕⊆ Im(A) (or equivalently, there is no (n − r)d-shrunksubspace). We want to construct a matrix A′′ ∈ X dd′ of rank larger than d′ rk(A).

There exists Bl ∈ X d and a vector vl ∈ Vl−1 such that Bl(vl) ∕∈ Im(A). We find amatrix Bl−1 and a vector vl−1 such that Bl−1(vl−1) = A(vl). Walking backwards, wefind matrices Bl−2, . . . , B1 and vectors vl−3, . . . , v1 such that vi ∈ Vi−1 and Bi−1(vi−1) =A(vi). As V0 = ker(A), we have A(v1) = 0.

Set A′ = A⊗ Id′ ∈ X dd′. As rk(A) = rd, we have rk(A⊗ Id′) = rdd′. Let Eij denotethe elementary matrices in Matd′,d′ so (i, j)-th entry of Eij is 1 and other entries are 0.Consider the matrix

B′ = B1 ⊗ E21 +B2 ⊗ E32 +B3 ⊗ E43 + · · ·+Bl−1 ⊗ El,l−1 +Bl ⊗ E1l.

If rk(B′) > rdd′, set A′′ = B′. Otherwise, set Y = 〈A′, B′〉 ⊆ Matndd′,ndd′ .

Let e1, . . . , ed′ be the standard basis vectors of Kd′ . Moreover, let

w1 = v1 ⊗ e1, w2 = v2 ⊗ e2, . . . , wl = vl ⊗ el ∈ Knd ⊗Kd′ .

Then, we have

A′w1 = (Av1)⊗ (Id′e1) = 0, A′wi = (Avi)⊗ (Id′ei) = (Bi−1vi−1)⊗ ei for i = 2, . . . , l.

Also we compute

B′wi = (Bivi)⊗ ei+1 = A′wi+1 for i < l, B′wl = (Blvl)⊗ e1.

In particular, ifZ0 ⊆ Z1 ⊆ · · · ⊆ Zk = Z∗

is the second Wong sequence of (A′,Y), then B′wi is contained in Zi. However, asB′wl = (Blvl)⊗e1 and Blvl ∕∈ Im(A), we deduce that B′wl ∕∈ Im(A′). Thus, Z∗ ∕⊆ Im(A′).By Lemma 3.4.21, we see that A′ is not of maximal rank in Y, so there exists µ ∈ K suchthat A′′ = A+µB has rank at least rdd′ +1. Note that the determinant of a submatrixof size rdd′ + 1× rdd′ + 1 of A+ µB has degree at most rdd′ + 1 in µ, thus we can findµ by evaluating this polynomial at rdd′ + 2 different points of K.

Step 3: We now use the constructive regularity lemma where input is A′′ which isof rank ≥ rdd′ + 1. Then the algorithm returns a matrix M of rank (r + 1)dd′ and thecorresponding full rank (r + 1)× (r + 1) submatrix.

We want to use the above theorem iteratively to obtain an algorithm for the compu-tation of the non-commutative rank. However, this requires some control over the sizeof d′: At each step, the size of the matrix spaces increase by a factor of d′ and we mayneed to apply the process n− 1 times. We use the following lemma to keep the sizes ofthe matrix spaces bounded by n+ 1:



Lemma 3.4.23 ([IQS15a], Lemma 5.2). Let X ⊆ Matn,n and d > n + 1. Assumethat we are given a matrix A ∈ X d of rank dn. Then there exists a deterministicpolynomial-time procedure that constructs A′ ∈ X d−1 of rank (d− 1)n.

Proof. Let A′′ be the (d−1)n× (d−1)n submatrix of A obtained by deleting kd-th rowsand kd-th columns for k = 1, . . . , n. Observe that we can identify A′′ with a matrix inX d−1. We claim that rk(A′′) > (d − 1)(n − 1): If not, as A′′ is obtained from A bydeleting n rows and n columns, we have

rk(A) ≤ rk(A′′) + 2n ≤ (d− 1)(n− 1) + 2n = dn+ n+ 1− d < dn

where the last inequality follows from the assumption d > n+1. This is a contradictionas rk(A) = dn so we deduce that rk(A′′) > (d−1)(n−1). Now we can use the constructiveregularity lemma to obtain a matrix A′ ∈ X d−1 of rank (d− 1)n.

Remark 3.4.24. Compare the above lemma with Corollary 3.3.7: There, we proved thatthe invariants of degree at most n(n − 1) cut out the nullcone of the left-right action.Using the Corollary 3.3.7, one can try to reduce the assumption d > n + 1 of theabove lemma to d > n − 1. Only problem is that the mentioned corollary by Derksenand Makam is not constructive. On the other hand, in [IQS15a], the authors give aconstructive version of the corollary and show that it is in fact possible to reduce theassumption of the above corollary. The interested reader may check [IQS15a], Section 6.

Using Theorem 3.4.22 together with the size reduction process described in the abovelemma, we are now ready to state the main algorithm of this section.

The main idea of the algorithm is to apply Theorem 3.4.22 iteratively: We initialized = 1 and start with an arbitrary matrix A ∈ X . Set r = rk(A). Then, we apply thetheorem to obtain one of the following:

1. An (n − r)-shrunk subspace . In this case, we are done. The non-commutativerank of X must be r as there exists a matrix of rank r in X and also there existsan (n− r)-shrunk subspace for X .

2. A matrix A′ ∈ X r+1 of rank at least (r + 1)2 with an (r + 1) × (r + 1)-blocksubmatrix of rank (r + 1)2.

Then, we set d = r + 1, replace A with the (r + 1) × (r + 1)-block matrix of A′ andrestart the process. If at some point the blow-up parameter d exceeds r + 1, we use theabove lemma to reduce it down to r + 1.



input : B1, . . . , Bm ∈ Matn,noutput : The non-commutative rank r of X = 〈B1, . . . , Bm〉 together with a matrix of rank

rd in X d for some d ≤ r + 1 as well as an (n− r)-shrunk subspace for Xinitialize: d = 1, A ∈ X an arbitrary matrix

1 while True do2 W0,W

∗ ← 03 l = 04 r = rk(A)5 d′ = r + 16 while l ≤ r + 1 and W ∗ ⊆ Im(A) do7 Vl+1 ← A−1(Wl)8 l ← l + 19 Wl ← XA−1(Wl−1)

10 W ∗ = Wl

11 end12 if W ∗ ⊆ Im(A) then13 return r as the non-commutative rank of X , A as the matrix of rank r in X and

A−1(W ∗) as the shrunk subspace14 end15 else16 Pick Bl ∈ X and vl ∈ Vl−1 such that Bl(vl) ∕∈ Im(A)17 i ← l − 118 while i > 0 do19 Pick Bi ∈ X and vi ∈ Vi−1 such that Bi(vi) = A(vi+1)20 i ← i− 1

21 end22 A′ ← A⊗ Id′

23 B′ ← B1 ⊗ E21 +B2 ⊗ E32 + · · ·+Bl ⊗ E1l

24 d ← d · d′25 if rk(B′) > rd then26 A′′ ← B′

27 end28 else29 Compute µ ∈ K such that A′ + µB′ has rank > rd30 A′′ ← A′ + µB′

31 end32 Use the constructive regularity lemma with input A′′ to obtain a matrix A∗ of rank

≥ (r + 1)d and an r + 1× r + 1-block submatrix M of A∗ of rank (r + 1)d33 A ← M34 if d > r + 1 then35 Use Lemma 3.4.23 repeatedly to obtain C ∈ X r+2 of rank (r + 1)(r + 2)36 A ← C

37 end

38 end

39 endAlgorithm 2: The computation of non-commutative rank of matrix subspaces



3.4.3 Orbit Closure Problem for the Simultaneous Conjugation

In this section, we give a polynomial time algorithm for the orbit closure intersectionproblem for simultaneous conjugation. Moreover, if the orbit closures do not intersectthen the algorithm successfully produces an invariant f ∈ K[Matmn,n]

GLn that separatesthe orbit closures.

Given matrices C1, . . . , Cm ∈ Matn,n, let C be the unital sub-algebra spanned byC1, . . . , Cm. In other words, C is the smallest subspace of Matn,n containing In, C1, . . . , Cm

that is closed under matrix multiplication.

In [DM18a], Derksen and Makam show the existence of a polynomial time algorithm toconstruct a basis for such algebras and use these bases to separate orbit closures.

Let [m]∗ be the set of words and to each word w = w1w2 . . . wk ∈ [m]∗, assign thematrix

Cw = Cw1Cw2 . . . Cwk

Also define Cε = In where ε denotes the empty word.

There is a total order on [m]∗ as follows:

Definition 3.4.25. Let w1 = i1i2 . . . ia and w2 = j1j2 . . . jb be given. We say w1 issmaller than w2 (and write w1 < w2) if either

1. a = l(w1) < l(w2) = b or

2. l(w1) = l(w2) and for the smallest integer k for which ik ∕= jk we have ik < jk.

Definition 3.4.26. A word w ∈ [m]∗ is called a pivot if

Cw ∕∈ 〈Cu | u < w〉

We denote the set of all pivot words by P .

Lemma 3.4.27. Let CP = Cw | w ∈ P. Then CP is a basis for C that is called thepivot basis.

Proof. We first need to show that CP spans C. Let w be the smallest word such thatCw ∕∈ 〈CP 〉. Then w is not a pivot as Cw ∕∈ CP . By the definition we have

Cw ∈ 〈Cu | u < w〉

and since w is the smallest word for which Cw ∕∈ 〈CP 〉, we have 〈Cu | u < w〉 ⊆ 〈CP 〉,which is a contradiction. Hence for each word w ∈ [m]∗, we have Cw ∈ 〈CP 〉 whichimplies that CP spans C.

Now assume that CP is linearly dependent, i.e. there exist pivots w1, . . . , wk ∈ [m]∗

and non-zero λ1, . . . ,λk ∈ K such that

λ1Cw1 + · · ·+ λkCwk= 0



Let wi be the largest word among w1, . . . , wk. Then

wi ∈ 〈w1, . . . , wi1 , wi+1, . . . , wk〉

which contradicts the fact that wi is a pivot. Thus, CP is linearly independent.

Given two words w1 = i1 . . . ia, w2 = j1 . . . jb, we define the concatenation of w1 andw2 as w1w2 = i1 . . . iaj1 . . . jb ∈ [m]∗.

Proposition 3.4.28. 1. If w is a non-pivot, then xwy is a non-pivot for all x, y ∈[m]∗.

2. Every sub-word of a pivot is a pivot.

3. For any non-empty word w ∈ [m]∗, wn is not a pivot.

4. Any word w containing a non-empty subword of the form un for some u ∈ [m]∗ isnot a pivot.

Proof. 1. Since w is a non-pivot, we can write

Cw =

k+

i=1

λkCwi

for some w1, . . . , wk < w. Then we also have

Cxwy = CxCwCy =

k+

i=1

λkCxCwiCy =

k+

i=1

λkCxwiy

Since wi < w we have xwiy < xwy and the result follows.

2. Let u be a sub-word of a pivot w. Then by definition of a sub-word, we can writew = xuy for some x, y ∈ [m]∗ and the result follows by the first part.

3. The Cayley-Hamilton theorem states that every matrix C satisfies its own charac-teristic polynomial (here, the characteristic polynomial is defined to be det(tI−C)).Therefore there exist λ0,λ1, . . . ,λn−1 ∈ K such that

Cn + λn−1Cn−1 + · · ·+ λ1C + λ0In = 0

For a non-empty word w ∈ [m]∗, we have Cwi = (Cw)i so by the Cayley-Hamilton

theorem Cwn is in the span of Cε, Cw, . . . , Cwn−1 and the result follows.

4. This part is a direct consequence of parts 2 and 3.

Proposition 3.4.29 ([DM18a]). Let w = w1w2 . . . wd ∈ [m]∗ be a word of length d andassume that w has no sub-words of the form un for u ∈ [m]∗ − ε. Then w has at least/d+12

0/n distinct sub-words.



Proof. Assume that u = wj+1 . . . wl and v = wi+1 . . . wk are two sub-words of w that areequal and j < i. If i ≤ l (alternatively, if 2i − j ≤ k), then wj+1 . . . wi is a sub-word ofu and hence, of v, so

wj+1 . . . wi = wi+1 . . . w2i−j

Similarly, if 2i− j ≤ l (3i− 2j ≤ k) we have

wi+1 . . . w2i−j = w2i−j+1 . . . w3i−2j

In general, the word wj+1 . . . wi is repeated ⌊(k − j)/(i− j)⌋ times. If i ≤ k/n then thesub-word appears

9k − i

i− j

:≥

9ni− j

i− j

:≥ n

which contradicts our assumption.

Now let S be the collection of all sub-words of w of the form wi+1 . . . wk such thati ≤ k/n. By the discussion above, all such words are necessarily distinct. Observe thatfor each k, there are ⌊k/n⌋+ 1 many choices for i thus w has at least

d+

k=0

9kn

:+ 1 >

d+

k=0

k

n=

/d+12

0

n

many sub-words.

Corollary 3.4.30. Let d be the length of the largest pivot. Then

d <√2n3

Proof. Since the set of pivots form a basis of C, the number of pivots equals dim C.Moreover, since each pivot satisfies the assumption of the above proposition and everysub-word of a pivot is a pivot, we have

/d+12

0

n≤ dim C

Moreover, as C is a subspace of Matn,n, dim C ≤ n2 so

d2

2n<

/d+12

0

n≤ dim C ≤ n2

which implies that d2 < 2n3 and the result follows.



Now, using the bound ≤√2n3, we can give a polynomial time algorithm to compute

a pivot basis:

input : C1, . . . , Cm ∈ Matn,noutput : The pivot basis P of the sub-algebra generated by C1, . . . , Cm

initialize: t = 1 and P = P0 = [(ε, In)]1 while Pt−1 ∕= [] do2 Pt = []3 foreach (w,Cw) ∈ Pt−1 do4 Pt = Pt + [(w1, CwC1), (w2, CwC2), . . . , (wm,CwCm)]5 end6 foreach (w,Cw) ∈ Pt do7 if w is a pivot then8 P = P + [(w,Cw)]9 end

10 else11 remove (w,Cw) from Pt

12 end

13 end14 t = t+ 1

15 end16 return P

Algorithm 3: Finding the pivot basis for a sub-algebra of Matn,n

Theorem 3.4.31. Algorithm 3 outputs a pivot basis P for the sub-algebra generated byC1, . . . , Cm using poly(m,n) many arithmetic operations.

Proof. In each step, we construct m|Pt−1| many elements for Pt. Since Pt−1 consistsof only pivots, it has at most n2 many elements. Thus, in each step we construct atmost mn2 many elements. Since the length of the largest pivot is bounded by

√2n3, we

construct polynomially many elements in total. Moreover, checking whether (w,Cw) isa pivot can be done in using poly(m,n) arithmetic operations as we only check if it isin the linear span of the elements constructed in previous steps.

In characteristic 0, we can immediately use the above algorithm for the orbit closureintersection problem for simultaneous conjugation:

Theorem 3.4.32. Algorithm 4 correctly outputs whether the orbit closures of A,B ∈Matmn,n intersect using poly(m,n) arithmetic operations under the assumption that char(K) =0.

Proof. Since GLn is a reductive group, the orbit closures of A and B intersect if andonly if f(A) = f(B) for all f ∈ K[Matmn,n]

GLn . By Procesi’s theorem (Theorem 3.1.4),the invariant ring is spanned by Trw.



input : A = (A1, . . . , Am), B = (B1, . . . , Bm) ∈ Matmn,noutput : Return YES if GLn ·A ∩GLn ·B ∕= ∅, a separating invariant

otherwise

initialize: Let Ci =

7Ai 00 Bi

8and C be the sub-algebra generated by Ci

1 Use Algorithm 3 to construct a pivot basis P for C2 foreach (w,Cw) ∈ P do

3 Cw =

7Aw 00 Bw

8

4 if Tr(Aw) ∕= Tr(Bw) then5 return Trw6 end

7 end8 return YESAlgorithm 4: Orbit Closure Intersection Problem for Simultaneous Conjugation

Since Tr is linear and Cw, w ∈ P is a linear basis of C, Trw(A) = Trw(B) for all w ∈ Pimplies that Trw(A) = Trw(B) for all w ∈ [m]∗ and the result follows.

Remark 3.4.33. For arbitrary characteristic, Procesi’s theorem does not hold so we needto use Donkin’s theorem. More precisely, in the fourth line of Algorithm 4, we need toreplace Tr(Aw) ∕= Tr(Bw) by σj(Aw) ∕= σj(Bw) where σj denotes the j-th characteristiccoefficient. However, since σj is not linear, it is not a priori clear why it is enough tocheck σj(Aw) ∕= σj(Bw) on a pivot basis. The following discussion shows why it actuallyworks.

Definition 3.4.34. Let K be an algebraically closed field and R be a finite dimensionalassociative K-algebra. A function N : R → K is called a norm if

1. N is a polynomial

2. N(1) = 1.

3. N(ab) = N(a)N(b) for all a, b ∈ R.

We state the following theorem without proof:

Theorem 3.4.35 ([DM18a]). Suppose that N1, N2 : R → K are two norms and a1, . . . , ak spanR as a K-vector space and N1(1 + tai) = N2(1 + tai) for all i and all t. Then we haveN1 = N2 on R.

Theorem 3.4.36. Algorithm 5 correctly outputs whether the orbit closures of A,B ∈Matmn,n intersect in polynomial time.



input : A = (A1, . . . , Am), B = (B1, . . . , Bm) ∈ Matmn,noutput : Return YES if GLn ·A ∩GLn ·B ∕= ∅, a separating invariant

otherwise

initialize: Let Ci =

7Ai 00 Bi

8and C be the sub-algebra generated by Ci

1 Use Algorithm 3 to construct a pivot basis P for C2 foreach (w,Cw) ∈ P do

3 Cw =

7Aw 00 Bw

8

4 if σj(Aw) ∕= σj(Bw) for some j = 1, . . . , n then5 return σj,w6 end

7 end8 return YESAlgorithm 5: Orbit Closure Intersection Problem for Simultaneous Conjugation

Proof. Define C to be the sub-algebra generated by Ci where

Ci =

7Ai 00 Bi

8

Define the norms N1, N2 on C where

N1(

7A 00 B

8) = detA, N2(

7A 00 B

8) = detB

Since the pivot basis P spans C, if σj(Aw) = σj(Bw) for all j = 1, . . . , n and w ∈ P ,we have det(In + tAw) ≡ det(In + tBw) so N1 ≡ N2 on C. In particular, we havedet(In + tAw) = det(In + tBw) for all w ∈ [m]∗ which implies by Donkin’s theorem thatthe orbit closures of A and B intersect.

Conversely, if the orbit closures of A and B intersect then σj(Aw) = σj(Bw) for allw ∈ [m]∗ so the fifth line of Algorithm 5 is never executed.

3.4.4 Orbit Closure Problem for the Left-Right Action

In this section, we give a polynomial time algorithm for the orbit closure intersectionproblem for the left-right action, defined as the action of SLn×SLn on Matmn,n via

(g, h) · (X1, . . . , Xm) = (gX1h−1, . . . , gXmh−1).

Moreover, once again we are able to give a separating invariant when the orbit closuresdo not intersect. First, we start with the following proposition that will use in thealgorithm.



Definition 3.4.37. If A = (A1, . . . , Am) ∈ Matmn,n, we define A[d] = (Ai ⊗ Ejk | i ∈[m], j, k ∈ [d]) ∈ Matmd2

nd,nd where (i, j, k) ∈ [m]× [d]× [d] is ordered lexicographically.

Proposition 3.4.38. Given A,B ∈ Matmn,n, the following are equivalent:

1. There exists f ∈ R(n,m) such that f(A) ∕= f(B),

2. There exists g ∈ R(nd,md2) such that g(A[d]) ∕= g(B[d]) for either d = n + 1 ord = n+ 2.

Proof. (⇒) As R(n,m) is spanned by fT for T ∈ Matme,e, we may without loss of gen-erality assume that f = fT for some T ∈ Matme,e and e ≥ 1. As f(A) ∕= f(B), eitherf(A) ∕= 0 or f(B) ∕= 0. Without loss of generality we assume that f(A) ∕= 0. Then wehave 1 ∕= f(B)/f(A) = µ ∈ K.

Observe that µn+1 and µn+2 cannot be simultaneously equal to 1 which implies thateither f(A)n+1 ∕= f(B)n+1 or f(A)n+2 ∕= f(B)n+2. Let d ∈ n+1, n+2 satisfy f(A)d ∕=f(B)d. To finish the proof, we need to find g ∈ R(nm, nd2) such that g(X [d]) = f(X)d.Consider

f(A)d = det# m+

i=1

Ai ⊗ Ti

$d

= det# m+

i=1

(

d+

k=1

(Ai ⊗ Ekk)⊗ Ti)$

= det#+

i,k

Ai ⊗ Ekk ⊗ Ti

$

= fS(A[d])

where S ∈ Matmd2e,e is given by Si,j,k = δj,kTi where δ denotes the Kronecker delta. Taking

g = fS , we get the result.

(⇒) Once again, we assume that g = fS for some S ∈ Matmd2e,e . Then

gS(A[d]) = det

#+

i,j,k

Si,j,k ⊗A[d]i,j,k

$

= det#+

i,j,k

Si,j,k ⊗Ai ⊗ Ejk

$

= det# m+

i=1

(+

j,k

Si,j,k ⊗ Ejk)⊗Ai

$

= fS′(A)

where S′ ∈ Matmed,ed is given by S′i =

,j,k Si,j,k ⊗ Ejk. Taking f = fS′ , the result

follows.



Corollary 3.4.39. The orbit closures of A and B do not intersect if and only if theorbit closures of A[d] and B[d] do not intersect for one choice of d ∈ n+ 1, n+ 2.

Remark 3.4.40. In [DM18a], the authors give the above proposition and the corollaryfor d ∈ n − 1, n with the intention of using the corollary in the main algorithm.The authors comment that for A ∈ Matmn,n, in the case that A is not in the nullcone,for any d ≥ n − 1 IQS algorithm produces an invariant fT for T ∈ Matmd,d such thatfT (A) ∕= 0. To our understanding, this requires a constructivization of Derksen andMakam’s results (showing γ(R(n,m)) ≤ n− 1) which is discussed in the last section of[IQS15a]. However, as we skipped this discussion, we use the original result by Ivanyos,Qiao and Subrahmanyam that produces invariants fT for d ≥ n+1 satisfying fT (A) ∕= 0.

The main idea of the algorithm is to reduce the problem to the orbit closure problemfor the simultaneous conjugation. In Section 3.3, we define the map

ϕ : Matmn,n → Matm+1n,n

(X1, . . . , Xm) )→ (I,X1, . . . , Xm)

and show that ϕ induces a surjection ϕ( : R(n,m + 1) → S(n,m). This immediatelyleads to the following propositions:

Proposition 3.4.41 ([DM18a]). Let A,B ∈ Matmn,n. Then we have

GLn ·A ∩GLn ·B ∕= ∅ ⇐⇒ SLn × SLn · ϕ(A) ∩ SLn × SLn · ϕ(B) ∕= ∅.

Proof. (⇒) Let F ∈ S(n,m) be a separating invariant for the orbit closures of A andB, i.e. F (A) ∕= F (B). Since ϕ( surjects R(n,m + 1) onto S(n,m), there exists f ∈R(n,m+ 1) such that ϕ(f = F . Then

f(ϕ(A)) = (ϕ(f)(A) = F (A) ∕= F (B) = f(ϕ(B))

so the orbit closures of ϕ(A) and ϕ(B) do not intersect.

(⇐) Let f ∈ R(n,m+ 1) be a separating invariant for ϕ(A) and ϕ(B). Then settingϕ(f = F ∈ S(n,m) we have

F (A) = f(ϕ(A)) ∕= f(ϕ(B)) = F (B)

so the orbit closures of A and B do not intersect.

Proposition 3.4.42 ([DM18a]). Let A = (A1, . . . , Am), B = (B1, . . . , Bm) ∈ Matmn,nsuch that det(A1) = det(B1) ∕= 0. Define

A = (A−11 A2, . . . , A

−11 Am), B = (B−1

1 B2, . . . , B−11 Bm)

Then we have

SLn × SLn ·A ∩ SLn × SLn ·B ∕= ∅ ⇐⇒ GLn · A ∩GLn · B ∕= ∅



Proof. Case 1: det(A1) = det(B1) = 1.

Let g = (A−11 , In). Then g · A = (In, A

−11 A2, . . . , A

−11 Am) = ϕ(A). Similarly for

h = (B−11 , In) we have h ·B = ϕ(B).

Now g · A and A have the same orbit and similarly h · B and B has the same orbit.Thus the orbit closures of A and B intersect if and only if the orbit closures of g ·A andh ·B intersect.

Since g ·A = ϕ(A) and h ·B = ϕ(B), using the above proposition we obtain the result.

Case 2: Let λ ∈ K be such that λn = det(A1). Set A′ = 1

λA and B′ = 1λB. Then the

orbit closures of A and B intersect if and only if the orbit closures of A′ and B′ intersect.

Moreover, A′ = A and B′ = B. Using Case 1, the orbit closures of A′ and B′ intersectif and only if the orbit closures of A′ and B′ intersect. We obtained the desired result.

Now we can give the algorithm. We will use Algorithm 2. Observe that givenA1, . . . , Am, Algorithm 2 returns

1. The non-commutative rank r of X = 〈A1, . . . , Am〉.

2. A matrix A ∈ X d of rank rd.

3. An (n− r)-shrunk subspace for X .

The algorithm can be used to test nullcone membership. A tuple (A1, . . . , Am) is in thenullcone if and only if ncrk(X ) < n. Moreover, if (A1, . . . , Am) is not in the nullcone,we can certify this with an invariant: The algorithm returns a matrix A ∈ X d of fullrank. We can efficiently find T1, . . . , Tm satisfying

A = X1 ⊗ T1 + · · ·+Xm ⊗ Tm.

Then, the polynomial

fT : Matmn,n → K

(A1, . . . , Am) )→ A1 ⊗ T1 + · · ·+Am ⊗ Tm

is an invariant satisfying fT (A1, . . . , Am) ∕= 0.

Moreover, we can produce such an invariant fT , T ∈ Matmd,d for any d ≥ n+ 1:

Lemma 3.4.43. Let A = (A1, . . . , Am) ∈ Matmn,n. If A is not in the nullcone of theleft-right action, then IQS algorithm can be used to produce an invariant fT , T ∈ Matmd,dsatisfying fT (A) ∕= 0 for any d ≥ n+ 1.

Proof. Observe that if IQS algorithm is called with input A, then it returns a matrixB in some certain blow-up 〈A1, . . . , Am〉e satisfying rk(B) = ne. Moreover, e satisfiese ≤ n + 1 by Lemma 3.4.23. Now assume that d ≥ n + 1 is given and let d′ satisfyed′ ≥ d. Then B⊗ Id′ ∈ 〈A1, . . . , Am〉ed′ satisfy rk(B) = ed′n. Now we can use Lemma



3.4.23 repeatedly to obtain B′ ∈ 〈A1, . . . , Am〉d satisfying rk(B′) = nd. Moreover, wecan efficiently expand

B′ =m+

i=1

+

j,k∈[d]λijkA1 ⊗ Ejk.

Setting

Ti =+

i,j,k

λijkEjk,

we have B′ = A1 ⊗ T1 + · · · + Am ⊗ Tm. Then, fT (A) = det(B′) ∕= 0. This invariant iswhat we wanted.

Theorem 3.4.44. Algorithm 6 correctly outputs whether the orbit closures of A,B ∈Matmn,n under the action of SLn × SLn intersect in polynomial time. Moreover, whenthe orbit closures do not intersect, the algorithm produces an invariant f ∈ R(n,m)saitsfying f(A) ∕= f(B).

Proof. In line 5, we check whether both A and B are in the nullcone using IQS algorithm.If both A and B are in the nullcone, then their orbit closures intersect so we return YES.In lines 8 and 13, we check whether only one of A and B is in the nullcone. If that isthe case, then the orbit closures do not intersect. In this case, we use the matrix of rankn(n+ 1) to produce an invariant fT that separates A and B.

If neither A nor B is in the nullcone, then using Lemma 3.4.43, we produce twoinvariants fT (n+1) and fT (n+2) satisfying fT (n+1)(A) ∕= 0, fT (n+2)(A) ∕= 0. If these twoinvariants satisfy

fT (n+1)(B) ∕= fT (n+2)(A) or fT (n+2)(B) ∕= fT (n+2)(A),

then the orbit closures do not intersect and we return the invariant. If both agree on Aand B, we proceed to the next step.

Now, we want to use Proposition 3.4.38. We need to check whether the orbit closuresof A[d] and B[d] intersect for d ∈ n+ 1, n+ 2. Observe that for d ∈ n+ 1, n+ 2, wehave

fT (d)(A) = det# m+

i=1

T (d)i ⊗Ai

$

= det# m+

i=1

(+

j,k∈[d](T (d)i)jkEjk)⊗Ai

$

= det#+

i,j,k

(T (d)i)jkAi ⊗ Ejk

$.

We set v(d) = ((T (d)i)jk | i ∈ [m], j, k ∈ [d]) and construct a matrix P (d) ∈ GLmd2 suchthat the first row of P (d) is v(d). Then, U(d) = P (d) ·A[d] satisfies

U(d)1 = (+

i,j,k

(T (d)i)jkAi ⊗ Ejk)



which implies that fT (d) = det(U(d)1). Similarly, if V (d) = P (d) ·B[d] then det(V (d)1) =

fT (d)(B). As the actions of SLnd × SLnd and GLmd2 on Matmd2

nd,nd commute, the orbit

closures of A[d] and B[d] intersect if and only if the orbit closures of U(d) and V (d)intersect. However, by Proposition 3.4.42, the orbit closures U(d) and V (d) intersect ifand only if the orbit closures of U(d) and V (d) intersect under simultaneous conjugation.

We can check this efficiently using Algorithm 5. Moreover, if the algorithm returns aninvariant f that separates U(d) and V (d), we can efficiently compute an invariant thatseparates A and B: Consider the polynomial

F (X1, . . . , Xmd2) = f(Adj(X1)X2, . . . ,Adj(X1)Xmd2).

Observe that F ∈ R(nm,md2) as

F (gA1h−1, . . . , gAmd2h

−1) = f(hAdj(A1)g−1gA2h

−1, . . . , hAdj(A1)g−1gAmd2h

−1)

= f(hAdj(A1)A2h−1, . . . , hAdj(A1)Amd2h

−1)

= f(Adj(A1)A2, . . . ,Adj(A1)Amd2)

= F (A1, . . . , Amd2), for (A1, . . . , Amd2) ∈ Matmd2

nd,nd .

Moreover, we have

F (U(d)) = f(Adj(U(d)1)U(d)2, . . . ,Adj(U(d)1)U(d)md2)

= det(U(d)1)deg(f)f(U(d)−1

1 U(d)2, . . . , U(d)−11 U(d)md2)

∕= det(V (d)1)deg(f)f(V (d)−1

1 V (d)2, . . . , V (d)−11 V (d)md2)

= F (V (d))

where the inequality follows by the fact that f separates U(d) and V (d) (and by the factthat det(U(d)1) = det(V (d)1)). Thus, F separates U(d) and V (d). As U(d) = P · A[d]

and V (d) = P ·B[d], we deduce that P−1 ·F separates A[d] and B[d]. Observe that P−1 ·Fcan be considered as an invariant in R(n,m) via the map

(A1, . . . , Am) )→ (P−1 · F )((Ai ⊗ Ejk | i ∈ [m], j, k[d]) = (P−1 · F )(A[d]).



input : A = (A1, . . . , Am), B = (B1, . . . , Bm) ∈ Matmn,noutput : Return YES if SLn × SLn ·A ∩ SLn × SLn ·B ∕= ∅, a separating

invariant otherwise1 Call Algorithm 2 with inputs A and B respectively2 rA, rB ← ncrk(〈A1, . . . , Am〉), ncrk(〈B1, . . . , Bm〉)3 X ← the n(n+1)× n(n+1)-matrix produced by the IQS algorithm with input A4 Y ← the n(n+1)× n(n+1)-matrix produced by the IQS algorithm with input B5 if rA < n and rB < n then6 return YES7 end8 else if rA = n but rB < n then9 Write Y =

,i,j,k λijkAi ⊗ Ejk

10 Ti ←,

j,k λijkEjk

11 return fT12 end13 else if rB = n but rA < n then14 Write Y =

,i,j,k λijkBi ⊗ Ejk

15 Ti ←,

j,k λijkEjk

16 return fT17 end18 else19 Compute fT (d) satisfying fT (d)(A) ∕= 0 for d ∈ n+ 1, n+ 2 (Lemma 3.4.43)

20 if fT (n+1)(B) ∕= fT (n+1)(A) or fT (n+2)(B) ∕= fT (n+2)(A) then

21 return fT (d) satisfying fT (d)(A) ∕= fT (d)(B)

22 end23 else24 foreach d ∈ n+ 1, n+ 2 do25 v = ((Ti)jk | i ∈ [m], j, k ∈ [d])26 P ← an invertible matrix such that its first row is v

27 U ← P ·A[d]

28 V ← P ·B[d]

29 U ← (U−11 U2, . . . , U

−11 Umd2)

30 V ← (V −11 V2, . . . , V

−11 Vmd2)

31 Use Algorithm 5 with input U and V

32 if the orbit closures of U and V do not intersect then

33 f ← the invariant that Algorithm 5 returns that separates U and V34 F (X1, . . . , Xmd2) ← f(Adj(X1)X2, . . . ,Adj(X1)Xmd2)35 return P−1 · F36 end

37 end38 return YES

39 end

40 endAlgorithm 6: Orbit Closure Intersection Problem for Left-Right Action


4 Invariant Theory of Quivers

In the previous chapter, we discussed the simultaneous conjugation and left-right ac-tion on the space of matrix tuples. The corresponding invariants are called the matrixinvariants and the matrix semi-invariants. The matrix invariants are the invariants ofendomorphisms of a vector space under a change of basis. Similarly, the matrix semi-invariants are the semi-invariants of linear maps from Kn to Kn under a change of basesof both spaces.

These two problems can be seen as particular examples to a more general theory ofquiver representations. Informally, a quiver is a directed graph and a quiver represen-tation is a collection of vector spaces on each vertex and linear maps for each arrow ofthe quiver. The group of invertible endomorphisms of each such vector space acts onthe quiver representation in an obvious way. As before, we define two types of invari-ants: the quiver invariants are the invariants of quiver representations under the actionof a product of general linear groups and the quiver semi-invariants are the invariantsof quiver representations under the action of a product of special linear groups. In thefirst case, we show that the trace function along each oriented cycle of the quiver givea generating set of invariants. In the second case, we assume that the quiver has nooriented cycle and show that in this case determinants of some certain block matricesspan the ring of semi-invariants.

In Section 4.1, we give a quick introduction to the basic objects and propositions ofquiver representations. The literature on the subject is quite wide, thus we only presentthe results that we will need for the remaining sections.

In Section 4.2, we study the ring of invariants of quiver representations. We presentLe Bruyn-Procesi theorem which states that in characteristic 0, the ring of invariantsis spanned by traces along oriented cycles. Donkin generalizes this result by replacingtrace function with characteristic coefficients. We present his proof in the end of thesection.

In Section 4.3, we focus on the semi-invariants of quivers. We first give a descriptionof the nullcone using King’s stability condition. Then in Section 4.3.2 we give a set ofgenerators for the ring of semi-invariants and give a determinantal description of thegenerators. Again, the section depends on the theory of matrix semi-invariants that wedevelop in previous chapter.

Lastly, in Section 4.4, we prove a degree bound for the set of generators for both theinvariant ring and the semi-invariant ring.


A more detailed treatment of this chapter (for K = C) can be found in [DW17].

As in the previous chapter, we fix an algebraically closed field K and assume everyvector space and coefficient is over K unless stated otherwise. Moreover, throughoutwe assume that all of our quivers are connected (as in the sense that the underlyingundirected graph is connected), though the most of the results can be trivially generalizedto arbitrary quivers.

4.1 Basic Definitions

A quiver is a directed graph. More formally, a quiver is a tuple Q = (Q0, Q1, h, t) whereQ0 is a finite set of vertices, Q1 is a finite set of arrows (directed edges) and h, t arefunctions

h, t : Q1 → Q0

where h(a) is called the head of the arrow a ∈ Q1 and t(a) is called the tail of the arrowa ∈ Q1. We drop the parentheses to simplify notation and write ha for h(a) and ta fort(a).

Definition 4.1.1. A representation V of a quiver Q is a collection (V (x) | x ∈ Q0) offinite dimensional vector spaces and a collection (V (a) | a ∈ Q1) of linear maps

V (a) : V (ta) → V (ha)

Given a representation V of Q, the dimension vector dim(V ) is the function

dim(V ) : Q0 → Z≥0

x )→ dim(V (x)).

We also call V a Q-representation.

For a fixed quiver Q, representations of Q will be the objects of the category of Q-representations. We define the morphisms as follows:

Definition 4.1.2. Let Q be a quiver and V,W be two representations of Q. Then acollection of linear maps φ = (φ(x) : V (x) → W (x) | x ∈ Q0) is called a morphismof quiver representations if for all a ∈ Q1 we have W (a)φ(ta) = φ(ha)V (a), i.e. thefollowing diagram

V (ta) V (ha)

W (ta) W (ha)

φ(ta)

V (a)

φ(ha)

W (a)

commutes. We denote the set of morphisms of quiver representations from V to W byHomQ(V,W ).



We denote the category of representations of Q by Rep(Q). The objects of Rep(Q) arerepresentations ofQ and morphisms of Rep(Q) are morphisms ofQ-representations.

Definition 4.1.3. Given a quiver Q and a dimension vector α : Q0 → Z≥0, we definethe representation space of α-dimensional representations as

Rep(Q,α) =!

a∈Q1

Matα(ha),α(ta) .

By identifying each vector space V (x) with Kα(x), we can identify α-dimensional repre-sentations with Rep(Q,α).

Let α be a dimension vector of the quiver Q. Set

GLα =!

x∈Q0

GLα(x).

We define an action of GLα on Rep(Q,α) as follows: Given (g(x) | x ∈ Q0) ∈ GLα andan α-dimensional representation (V (a) | a ∈ Q1) ∈ Rep(Q,α) we set

(g(x) | x ∈ Q0) · (V (a) | a ∈ Q1) = (g(ha)V (a)g(ta)−1 | a ∈ Q1).

Observe that this action corresponds to a change of bases of each V (x), x ∈ Q0. More-over, g ∈ GLα is a morphism of Q-representations from V ∈ Rep(Q,α) to g · V ∈Rep(Q,α).

Example 4.1.4. Let Q be the quiver

am

a2a1

which we call the m-loop quiver. For the dimension vector α = (n) we have GLα = GLn

and Rep(Q,α) = Matmn,n. The action is given by

g · (X1, . . . , Xm) = (gXg−1, . . . , gXmg−1),

i.e. simultaneous conjugation. Similarly let Q be the quiver

...

a1

am

which we call the generalized Kronecker quiver and denote it by θ(m). For dimensionvector α = (n, n) we have GLα = GLn × GLn and Rep(Q,α) = Matmn,n. The action isgiven by

(h, g) · (X1, . . . , Xm) = (gX1h−1, . . . , gXmh−1)

i.e. the left-right action.



The above examples formally shows a fact that we implied in the introduction ofthis chapter: The quiver representations generalize simultaneous conjugation and theleft-right action that are discussed in the previous chapter.

Definition 4.1.5. Let V ∈ Rep(Q,α). A sub-representation W of V is a collection ofsubspaces

W (x) ⊆ V (x), x ∈ Q0

with the property that V (a)(W (ta)) ⊆ W (ha) for all a ∈ Q1. In this case, W is also arepresentation of Q with the collection of linear maps

W (a) = V (a) |W (a), a ∈ Q1.

We also call an injective morphism

φ : W → V

a sub-representation of V as φ induces an isomorphism betweenW and a sub-representationW ′ of V where W ′(x) = φ(x)(W (x)).

Example 4.1.6. Let Q = θ(m) be the generalized Kronecker quiver with m arrows. Inthe previous example we have shown that Rep(Q, (n, n)) = Matmn,n where the action isgiven by the left-right action. Given a representation V = (X1, . . . , Xm) ∈ Matmn,n, apair (U,W ) of subspaces of U,W ⊆ Kn, induces a sub-representation of V if and only ifXi(U) ⊆ V for all i = 1, . . . ,m.

Definition 4.1.7. Given a quiver Q, the path algebra KQ is defined to be the vectorspace with basis

eq | q is a path in Q

(here we also assume that for each vertex x ∈ Q0, there exists a path of length 0 withthe initial and final vertices are x. By an abuse of notation, we denote this path withex) such that there is a multiplication given by

ep · eq =;0 if the final vertex of p is not the initial vertex of q

epq otherwise

on the basis. Here, pq denotes the concatenation of the paths p and q.

Observe that +

x∈Q0

ex

is the unit of the algebra KQ.

Proposition 4.1.8. The category Rep(Q) of quiver representations and the categoryKQ-mod of finitely generated KQ-modules are equivalent.

Proof. For the proof, see [DW17], Proposition 1.5.4.



Remark 4.1.9. The above proposition shows that Rep(Q) is an abelian category. Inparticular, every morphism of quiver representations has a kernel and a cokernel.

Definition 4.1.10. Let V and W be representations of Q. We define the map dVW tobe

<x∈Q0

HomK(V (x),W (x))<

a∈Q1HomK(V (ta),W (ha))

dVW

where dVW (φ(x) | x ∈ Q0) = (φ(ha)V (a)−W (a)φ(ta) | a ∈ Q1).

Remark 4.1.11. There is an obvious inclusion of HomQ(V,W ) into<

x∈Q0HomK(V (x),W (x))

since by definition a Q-morphism φ : V → W is a collection of linear maps φ(x) : V (x) →W (x) satisfying some certain conditions. In fact, it should be clear that the conditionis nothing but dVW (φ) = 0, i.e. the kernel of dVW is exactly HomQ(V,W ).

Since Rep(Q) is an abelian category, dVW also has a cokernel and we will denote itby ExtQ(V,W ). We remark that in homological algebra, the notation Ext is used forthe Ext-functor, however there should be no confusion since ExtQ(V,W ) actually is theExt-functor for the category Rep(Q). As we do not use this fact later, we will not definewhat Ext-functor is and we will use the notation ExtQ(V,W ) only to denote the cokernelof dVW . Thus, the following definition can also be interpreted as a proposition.

Definition 4.1.12. Let ExtQ(V,W ) denote the cokernel of dVW , i.e. there exists an exactsequence

0 HomQ(V,W )<

x∈Q0HomK(V (x),W (x))

<a∈Q1

HomK(V (ta),W (ha)) ExtQ(V,W ) 0.dVW

If α is the dimension vector of V and β is the dimension vector of W , then

dim# "

x∈Q0

HomK(V (x),W (x))$=

+

x∈Q0

α(x)β(x).

Similarly

dim# "

a∈Q1

HomK(V (ta),W (ha))$=

+

a∈Q1

α(ha)β(ta)

Definition 4.1.13. We define 〈α,β〉 to be the difference

〈α,β〉 =+

x∈Q0

α(x)β(x)−+

a∈Q1

α(ha).β(ta) = dimHomQ(V,W )− dimExtQ(V,W )

In particular, 〈α,β〉 = 0 if and only if dVW is a square matrix.



Definition 4.1.14. Let Q be a quiver without oriented cycles and let x ∈ Q0. We definea representation Px of Q as follows: For each y ∈ Q0, we define the vector space Px(y)having the basis [x, y] where [x, y] denotes the set of all distinct paths from x to y. Here,if y = x we define [x, x] to be the set with one element corresponding to the empty pathex. For each arrow a ∈ Q1, we define the map

Px(a) : [x, ta] → [x, ha]

p )→ ap

and linearly extend to a linear map Px(a) : Px(ta) → Px(ha).

Proposition 4.1.15. Let Q be a quiver without oriented cycles. Then

1. Px are projective objects in the category of Q-representations, i.e. for any epimor-phism of Q-representations f : V → V ′ and for every morphism g : P → V ′ ofQ-representations, there exists a Q-morphism h : P → V such that fh = g.

2. The indecomposable projective objects are all of the form Px for x ∈ Q0.

3. For any Q-representation V , HomQ(Px, V ) and V (x) are canonically isomorphic.

Proof. For the first two parts, we refer to [DW17], Corollary 2.1.4, Lemma 2.2.1 andProposition 2.2.3.

For the last part, define

ψ : HomQ(Px, V ) → V (x)

f )→ f(x)(ex)

where f(x) denotes the linear map f(x) : Px(x) = K · ex → V (x) and ex denotes theempty path from x to x. Conversely, we define

φ : V (x) → HomQ(Px, V )

v )→ fv

where fv is the morphism fv(y) : Px(y) → V (y) is given by fv(p) = V (p)v for each pathp ∈ [x, y].

4.2 Invariant Theory of Quivers

In this section, we prove Le Bruyn-Procesi theorem which states that the invariant ringof quiver representations is spanned by the traces along oriented cycles in characteristic0. Moreover, we show that in positive characteristic it is enough to replace traces withcharacteristic coefficients to generate ring of invariants.



4.2.1 Le Bruyn-Procesi Theorem

In Section 3.1.2, we proved that the ring of matrix invariants S(n,m) is generated bytraces of words of matrices. Recall that in Example 4.1.4, we discussed how the quiverrepresentations generalize the simultaneous conjugation using the m-loop quiver. In thatsense, Le Bruyn-Procesi theorem is the generalization of Procesi’s theorem to arbitraryquivers.

Definition 4.2.1. Let Q be a quiver and p = ak . . . a2a1 be an oriented path. We definea linear map V (p) : V (ta1) → V (hak) given by the product

V (p) = V (ak)V (ak−1) . . . V (a2)V (a1).

Observe that V (p) is well-defined since p is a path: For all s, we have tas+1 = has, sothe number of columns of V (as+1) and the number of rows of V (as) are equal. When pis a cycle, V (p) is an endomorphism and for g ∈ GLα we have

(g · V )(p) = g(hak)V (ak)g(tak)−1g(hak−1)V (ak−1) . . . V (a2)g(ta2)g(ha1)V (a1)g(ta1)

−1

= g(ta1)V (p)g(ta1)−1.

Thus, Tr(V (p)) is an invariant. We will call this invariant trace along the oriented cyclep.

Example 4.2.2. Consider the quiver

Q :

a1 a2

a3

Let’s fix the dimension vector α = (n, n). Then Rep(Q,α) = Mat3n,n and in coordinatesthe action is given as follows:

(g, h) · (A,B,C) = (gAg−1, hBg−1, gCh−1)

where g, h ∈ GLn, A is the linear map corresponding to the arrow a1, B is the linearmap corresponding to a2 and C is the linear map corresponding to a3.

Observe that for all m, Tr(Am) and Tr((BC)m) are invariants. These invariants arespecial cases of what we call the trace along an oriented cycle : In Q, the path a1a1 . . . a1is an oriented cycle and similarly a3a2a3a2 . . . a3a2 is an oriented cycle. To these cycles,we assign the linear maps Am and (BC)m. Then, their traces are invariants by the abovediscussion.

Now we state Le Bruyn-Procesi Theorem. We follow the proof given in [DW17]:



Theorem 4.2.3 ([LP90]). Assume that char(K) = 0. Let Q be a quiver and let α bea dimension vector. Then the invariant ring K[Rep(Q,α)]GLα is spanned by the tracesalong oriented cycles.

Proof. Let N = |Q0| be the number of vertices. We will proceed by induction on n.

For N = 1, the action is nothing but simultaneous conjugation. In other words, ifthere are m loops on the vertex and if X1, . . . , Xm are the linear maps corresponding toeach loop, then the action of GLα on Rep(Q,α) is

g · (X1, . . . , Xm) = (gX1g−1, . . . , gXmg−1).

where g ∈ GLα and Xi ∈ Matα,α. By Procesi’s theorem (Theorem 3.1.4), the invariantsof this action is given by the traces of words inXi. Since each word can also be consideredas an oriented cycle, we obtain the result for N = 1.

Now assume that the result holds for |Q0| = 1, . . . , N − 1. Let x ∈ Q0. Given arepresentation V ∈ Rep(Q,α), we define 3 spaces:

X ="

a∈Q1,ha ∕=x ∕=ta

Hom(V (ta), V (ha)),

W ="

a∈Q1,ha=x

V (ta)

Z ="

a∈Q1,ta=x

V (ha).

Assume that there arem loops on x. By abuse of notation, letGL(x) denoteGL(V (x)).Then as a representation of GL(x), Rep(Q,α) splits as

Rep(Q,α) ∼= Hom(V (x), V (x))m ⊕Hom(W,V (x))⊕Hom(V (x), Z)⊕X

∼= Hom(V (x), V (x))m ⊕ V (x)dimW ⊕ (V (x)∗)dimZ ⊕KdimX .

where the last isomorphism follows by the fact that GL(x) acts trivially on X, W andZ.

Using full version of Procesi’s theorem (Theorem 3.1.7), the invariants of this lastspace is spanned by

1. Trw(Y1, . . . , Ym) for w ∈ [m]∗,

2. ϕj(Ywvi) for w ∈ [m]∗

where (Y1, . . . , Ym, v1, . . . , vp,ϕ1, . . . ,ϕq) ∈ Hom(V (x), V (x))m⊕V (x)dimW⊕(V (x)∗)dimZ ,and lastly

3. K[X] since GL(x) acts trivially on X.



Observe that the invariants of the form given in 1 are in one-to-one correspondencewith the traces of oriented cycles where each arrow is a loop on the vertex x.

The invariants of the form given in 2 are the entries of V (q) for a path q of the form

q = a1a2 :x

a1 a2

and lastly, the invariants of the form given in 3 are the entries of V (q) for a path q thatdoes not pass through x.

Since K[Rep(Q,α)]GL(x) is finitely generated, there exist cycles p1, . . . , pr and pathsq1, . . . , qs such that the invariant ring spanned by

Tr(V (pi)) | i = 1, . . . , r ∪ V (qi)jk | i = 1, . . . , s

where V (qi)jk denotes the (j, k)-th entry of V (qi).

We define a new quiver as follows: Let Q′0 = Q0 − x and Q′

1 = b1, . . . , bs be aset of arrows such that hbi and tbi are the final and the initial vertices of the path qirespectively. Moreover, set α′ = α |Q′

0. In other words, we delete the vertex x and

replace each path qi with an arrow bi. Consider the map

π : Rep(Q,α) → Rep(Q′,α′)⊕Kr

V )→ (V (q1), . . . , V (qs),Tr(V (p1)), . . . ,Tr(V (pr))).

Observe that since the invariant ring is spanned by the entries of V (qi) and Tr(V (pi)),π is the categorical quotient map and we have

π((K[Rep(Q′,α′)⊕Kr]) = K[Rep(Q,α)]GL(x).

Moreover, π is a GLα′ equivariant map where the action of GLα′ on Kr is assumed tobe trivial. Thus

π((K[Rep(Q′,α′)⊕Kr]GLα′ ) = (K[Rep(Q,α)]GL(x))GLα′ = K[Rep(Q,α)]GLα .

NowK[Rep(Q′,α′)⊕Kr]GLα′ = K[Rep(Q′,α′)]GLα′⊗K[z1, . . . , zr] and by induction weknow that K[Rep(Q′,α′)]GLα′ is generated by traces along oriented cycles. If bi1bi2 . . . bikis an oriented cycle of Q′, then qi1qi2 . . . qik is an oriented cycle of Q and moreover

π(Tr(π(V )(bi1 . . . bik)) = Tr(V (qi1 . . . qik)).

Moreover, for a polynomial f(z1, . . . , zr) ∈ K[z1, . . . , zr] we have

π(f = f(Tr(V (p1)), . . . ,Tr(V (pr))) ∈ K[V (p1), . . . , V (pr)]

We deduce that the invariant ring K[Rep(Q,α]GLα is generated by Tr(V (qi1 . . . qik)) andTr(V (pi)), which are all traces along oriented cycles.



4.2.2 Donkin’s Theorem

Recall Section 3.1: In order to generate the ring of matrix invariants S(n,m) in positivecharacteristic, one needs to replace traces of words of matrices by the characteristiccoefficients of words of matrices. The proof of Le Bruyn-Procesi theorem suggests thata similar phenomena should occur in quiver representations. In fact, this is the case asDonkin proves:

Theorem 4.2.4 ([Don94]). The invariant ring K[Rep(Q,α)]GLα is generated by thecharacteristic coefficients along oriented cycles.

Proof. We prove the theorem in two steps. In the first step we assume that for any twovertices x, y ∈ Q0, there are exactly m arrows that have x as the tail and y as the head.We prove the theorem in this case.

In the second step, we will complete Q to the case considered in the first step andshow that if the theorem holds for the completion of Q, it must hold for Q.

Step 1 : Assume that for all pairs x, y ∈ Q0, there exist exactly m arrows having x asthe tail and y as the head.

SetV =

"

x∈Q0

Kα(x).

Then, we have a GLα-module isomorphism

Rep(Q,α) ="

a∈Q1

Hom(Kα(ta),Kα(ha)) =

m"

i=1

"

x,y∈Q0

Hom(Kα(x),Kα(y)) ∼= Hom(V, V )m.

Here, the isomorphism in the last step is given by the map<

x,y∈Q0Hom(Kα(x),Kα(y)) →

Hom(V, V ) with

(fx,y | x, y ∈ Q0) )→

%

&&&&'

fx1,x1 fx1,x2 . . . fx1,xk

fx2,x1

. . ....

.... . .

...fxk,x1 . . . . . . fxk,xk

(

))))*.

Observe that GLα is naturally a subgroup of GL(V ), given by the action of GLα

on V . More concretely, GLα can be identified with the set of matrices in GL(V )of the block diagonal form

%

&&&'

g(x1)g(x2)

. . .

g(xk)

(

)))*



where x1, . . . , xk are the vertices of Q and each g(xi) is an invertible α(xi)×α(xi)-matrix.

Let λ(x), x ∈ Q0 be distinct scalars and let

A =

%

&&&'

λ(x1)Iα(x1)

λ(x2)Iα(x2)

. . .

λ(xk)Iα(xk)

(

)))*.

With the identification of GLα as a subgroup of GL(V ) above, we have

GLα = CGL(V )(A) = g ∈ GL(V ) | gA = Ag.

Now, by Theorem 3.1.16, K[Hom(V, V )m]GLα is spanned by the invariants of theform

σj(Aq1Xw1A

q2Xw2 . . . AqsXws)

where w ∈ [m]∗, each Xi is an endomorphism of V and q1, . . . , qk ≥ 0.

The isomorphism K[Rep(Q,α)] ∼= K[Hom(V, V )m] maps this invariant to the func-tion

(f ix,y | x, y ∈ Q0, i = 1, . . . ,m) )→ σj

3%

&&&&'

λ(x1)q1fw1

x1,x1λ(x1)

q1fw1x1,x2

. . . λ(x1)q1fw1

x1,xk

λ(x2)q1fw1

x2,x1

. . ....

.... . .

...λ(xk)

q1fw1xk,x1

. . . . . . λ(xk)q1fw1

xk,xk

(

))))*

. . .

%

&&&&'

λ(x1)qsfws

x1,x1λ(x1)

qsfwsx1,x2

. . . λ(x1)qsfws

x1,xk

λ(x2)qsfws

x2,x1

. . ....

.... . .

...λ(xk)

qsfwsxk,x1

. . . . . . λ(xk)qsfws

xk,xk

(

))))*

4

(4.1)

By direct computation, the (p, q)-th block of the matrix inside σj is a linear com-bination of the matrices of the form

fw1xv1 ,xv2

fw2xv2 ,xv3

. . . fwsxvs ,xvs+1

.

where v1 = p and vk+1 = q.

Thus, we can write this matrix as a linear combination of block matrices havingfw1xv1 ,xv2

fw2xv2 ,xv3

. . . fwsxvs ,xvs+1

as its (p, q)-th entry and 0 everywhere else.

Now, for a tuple (X1, . . . , Xm) ∈ Matmn,n and a linear combinationX ofX1, . . . , Xm,the map σj(X) is an invariant of (X1, . . . , Xm). Thus, by Donkin’s theorem for



matrix invariants (Theorem 3.1.15), σj(X) is in the space spanned by σi(Xw) fori = 1, . . . , n and w ∈ [m]∗.

Applying this fact to the matrix above, we deduce that the invariant in (4.1), is inthe space spanned by σi(M) where M is a block matrix having

fw1xv1 ,xv2

fw2xv2 ,xv3

. . . fwsxvs ,xvs+1

as its (p, q)-th entry and 0 everywhere else. However, if p ∕= q, then the matrixhave 0 as block diagonals and thus its characteristic coefficient is 0. If p = q,then the given path xv1xv2 . . . xvk+1

is an oriented cycle. Thus, the invariant ringK[Rep(Q,α)]GLα is spanned by the characteristic coefficients along oriented cycles.

Step 2 : Let m = max2a ∈ Q1 | ta = x, ha = y | x, y ∈ Q0 be the maximumnumber of arrows between two vertices of Q. Extend Q to the quiver P by addingarrows between each pair of vertices x, y such that there are exactly m arrows fromx to y. Also, let R be the quiver obtained by deleting each arrow of Q′ that alsobelongs to Q. There is an isomorphism

Rep(P,α) = Rep(Q,α)⊕ Rep(R,α).

Here, the inclusion Rep(Q,α) → Rep(P,α) is given by adding zero maps for eacharrow a ∈ Q1 that does not belong to P . The same holds for the inclusionRep(R,α) → Rep(P,α). The induced ring map K[Rep(P,α)] → K[Rep(Q,α)]gives a surjection

K[Rep(P,α)]GLα → K[Rep(Q,α)]GLα .

By the previous step, the invariant ring on the left is spanned by elements of theform

σj(V (a1)V (a2) . . . V (ak))

where a1a2 . . . ak is an oriented cycle of P . The images of these polynomials spanthe invariant ring K[Rep(Q,α)]GLα . On the other hand, if ai is not in the quiverQ for some arrow ai, then this invariant maps to 0, and if all arrows ai are in Q,then the invariant maps to itself. Thus, the invariants of Rep(Q,α) are spannedby characteristic coefficients along oriented cycles.

Donkin’s theorem has the following interesting consequence:

Corollary 4.2.5 ([DM16b]). Let Q be a quiver and α be a dimension vector. There isa surjective degree 1 ring map

ψ( : S(N,M) → K[Rep(Q,α)]GLα

where M = |Q1| and N =,

x∈Q0α(x).



Proof. Set V =<

x∈Q0Kα(x). Note that dimV = N . Consider the injection

ψ : Rep(Q,α) → Hom(V, V )M

W = (W (a) | a ∈ Q1) )→ (F (a) | a ∈ Q1)

where F (a) is the |Q0| × |Q0|-block matrix having W (a) in the entry corresponding to(ta, ha) and 0 everywhere else. Then ψ is linear and its induced ring map ψ( is a degree1 ring map. We want to show that ψ((S(N,M)) = K[Rep(Q,α)]GLα . Let a1a2 . . . as bean oriented cycle in Q. Consider the word w = asas−1 . . . a2a1. Then,

ψ((σj,w)(W ) =

;0 if j > α(ta1)

σj(W (as) . . .W (a1)) otherwise

We obtained the desired result.

4.3 Semi-Invariants of Quivers

In the previous section, we discussed the invariants of quiver representations and showthat the ring of invariants is generated by traces (or characteristic coefficients) alongoriented cycles. Therefore, if Q is a quiver with no oriented cycles, the only invariantsare constants. However, the ring of semi-invariants can still be interesting.

Recall that for a dimension vector α of a quiver with n = |Q0| vertices, the groupcharacters of GLα are of the form

σ = (σ1, . . . ,σn) : GLα → K

(g1, . . . , gn) )→n!

i=1

det(gi)σi

for σ ∈ Zn. Thus, a semi-invariant f ∈ K[Rep(Q,α)] is a polynomial that satisfies

(g1, . . . , gn) · f = σ(g1, . . . , gn) · f, ∀g = (g1, . . . , gn) ∈ GLα

for some group character σ ∈ Zn. We define SI(Q,α) to be the ring of semi-invariants.We have a decomposition of SI(Q,α) with respect to the group characters:

SI(Q,α) ="

σ∈Zn

SI(Q,α)σ

where SI(Q,α)σ is the subspace of semi-invariants corresponding to the character σ. Asin the case of matrix semi-invariants, we have

SI(Q,α) = K[Rep(Q,α)]SLα ,

i.e. the semi-invariants for GLα are the invariants with respect to SLα.



In this section, we start by giving a description of the nullcone of quiver representationsunder the action of SLα using King’s stability condition. Then we turn our attentionto finding a set of generators. Under the assumption that the quiver has no orientedcycles, they are given by Schofield semi-invariants that we discuss in Section 4.3.2. Inthe end of the section, we give degree bounds.

4.3.1 King’s Stability Condition

Recall that given a group G, a rational representation V and a group character σ : G →K, a semi-stable (respectively stable) vector v is called σ-semi-stable (respectively σ-stable) if there exists a semi-invariant f of weight dσ for some d such that f(v) ∕= 0 (seeDefinition 2.2.36). King’s stability condition gives the conditions for a quiver represen-tation to be σ-semi-stable in terms of some condition on its sub-representations.

Theorem 4.3.1 ([Kin94]). Let Q be a quiver and V ∈ Rep(Q,α) be a representation ofQ of dimension α. Given a group character σ we have

1. V is σ-semi-stable if and only if

σ · α =+

x∈Q0

σ(x)α(x) = 0

and for every sub-representation W of V of dimension β we have σ · β ≤ 0.

2. V is σ-stable if and only if σ·α = 0 and for every non-zero, proper sub-representationW of V of dimension β we have σ · β < 0.

Proof. We use the Hilbert-Mumford criterion (see Theorem 2.2.38).

1. (⇒) Let V be σ-semi-stable. For each β : Q0 → Z with β(x) ≤ α(x) for all x ∈ Q0,we define

λβ : K× → GLα

t )→3

%

&&&&&&&&&&'

tt

. . .

t1

. . .

1

(

))))))))))*

β(x)

α(x)−β(x)

===== x ∈ Q0

4.

Observe that λα acts trivially on Rep(Q,α) as

(λα(t) · V )(a) = (t · Iα(ha)) · V (a) · (t−1 · Iα(ta)) = V (a)



Let W be a sub-representation of V of dimension β. Pick bases of V (x) for eachx ∈ Q0 such that the first dimW (x) vectors span W (x). Then with respect to thisbasis, V (a) is a block matrix of the form

V (a) =

7A B0 C

8, a ∈ Q1

where A is a dimW (ha) × dimW (ta)-matrix. Note that we have a zero block inthe bottom left corner since W is a sub-representation, i.e. V (a) maps W (ta) toW (ha). Thus,

(λβ(t) · V )(a) =#λ(t)(ha)

$V (a)

#λ(t)(ta)−1

$

=

7A t ·B0 C

8

so limt→0 λβ(t) · V exists. In other words, λα acts trivially on Rep(Q,α) and ifthere exists a β-dimensional sub-representation of V , then the limit limt→0 λβ(t)·Vexists. Using the Hilbert-Mumford criterion (Theorem 2.2.38), we get

〈λα,σ〉 = 0

and for each sub-representation of dimension β,

〈λβ ,σ〉 ≤ 0.

Here, 〈λβ ,σ〉 denotes the pairing of one-parameter subgroups and group characters.We compute the pairing:

σ λβ : K× → K×

t )→!

x∈Q0

det(λβ(t)(x))σ(x)

=!

x∈Q0

tβ(x)σ(x)

= tσ·β

so σ · β = 〈λβ ,σ〉 ≤ 0 and σ · α = 〈λα,σ〉 = 0.

(⇐) Let σ be as in the assumption and let λ : K× → GLα be a one-parametersubgroup such that limt→0 λ(t) · V exists. We need to show that 〈λ,σ〉 ≤ 0. Thereis a decomposition

V (x) ="

n∈ZW (n)(x)



where λ acts on W (n)(x) via multiplication by tn. In coordinates, this correspondsto choosing bases for each V (x) such that λ(t)(x) is the diagonal matrix

λ(t)(x) =

%

&&&&&&&&&&&&&&&&&&'

tn1

. . .

tn1

tn2

. . .

tn2

. . .

tnr

. . .

tnr

(

))))))))))))))))))*

and setting W (n)(x) = 〈ej | λ(t)(x)jj = tn〉.

For a ∈ Q1, observe that V (a)ij is multiplied by tn−m where ei ∈ W (n)(ha) andej ∈ W (m)(ta). Since limt→0 λ(t) · V exists, for each i, j with n−m ≤ 0, we musthave V (a)ij = 0. We deduce that V (a) maps W (n)(ta) to

<m≤nW

(m)(ha).

For fixed n, set W (≤n)(x) =<

m≤nW(m)(x) for each x ∈ Q0. Then by the above

discussion, W (≤n) is a sub-representation of V . For large enough n, this sub-representation is V and for small enough n it is 0. By the assumption we haveσ · α = 0 and for all k, σ · dim(W (≤k)) ≤ 0. But

〈λ,σ〉 =+

x∈Q0

σ(x)+

k∈Zk dimW (k)(x)

=+

k∈Zk+

x∈Q0

σ(x) dimW (k)(x)

=+

k∈Zk#σ · dim(W (≤k)/W (≤k−1))

$

=+

k∈Zσ · dimW (≤k) ≤ 0.

Here, all sums are finite since for large enough k and for small enough k we have

σ · dimW (≤k) = 0.

We obtained the desired inequality 〈λ,σ〉 ≤ 0 and the result follows by the Hilbert-Mumford criterion.

2. The proof goes similar to the first part when we use the second item of Theorem2.2.38.



Remark 4.3.2. Assume that V ∈ Rep(Q,α) is a σ-semi-stable representation and W isa sub-representation of V satisfying σ · dim(W ) = 0. Then, each sub-representation Uof W is also a sub-representation of V , thus it satisfies σ · U ≤ 0. Hence, W is alsoσ-semi-stable by King’s stability condition. Similarly, if U is a sub-representation ofV/W , then there is a sub-representation U of V containing W such that U/W = U .Observe that

σ · dim(U) = σ · dim(U/W ) = σ · dim(U)− σ · dim(W ) = σ · dim(U) ≤ 0

which implies that V/W is also σ-semi-stable. If we continue this process of pickingsub-representations with dimension vectors orthogonal to σ, we end up at a filtration

0 = V0 # V1 # · · · # Vk = V

of V such that each factor Vi+1/Vi is σ-stable.

Using King’s stability condition, we can characterize the nullcone of the action of SLα

on Rep(Q,α).

Corollary 4.3.3. Let α be a dimension vector. A representation V ∈ Rep(Q,α) is inthe nullcone of the action of SLα on Rep(Q,α) if and only if for all 0 ∕= σ ∈ ZQ0 withσ · dim(V ) = 0 there exists a sub-representation W of V satisfying

σ · dim(W ) > 0.

Proof. (⇒) Assume that V is in the nullcone and let σ ∈ ZQ0 . Since V is in thenullcone, V is not σ-semi-stable which implies that either σ · dim(V ) ∕= 0 or there existsa sub-representation W of V satisfying σ · dim(W ) > 0.

(⇐) Let σ ∈ ZQ0 . If σ · dim(V ) ∕= 0, then there are no non-zero semi-invariantsof weight σ. Thus we need to show that V is not σ-semi-stable for each σ satisfyingσ · dim(V ) = 0. By the assumption, for each such σ there exists a sub-representation Wsuch that

σ · dim(W ) > 0.

By the theorem, V is not σ-semi-stable and the result follows.

Example 4.3.4. Let Q be the generalized Kronecker quiver θ(m). We label the verticesof Q as 1 and 2 and assume the tail of each arrow is 1.

Let α = (q, p). Then

σ ∈ Z2 | σ · α = 0 = (−qk, pk) | k ∈ Z.

Using the above corollary, a representation V ∈ Rep(Q,α) is in the nullcone if and onlyif there exists a sub-representation W with the property that

q dimW (1)− p dimW (2) > 0.



Compare this fact with Theorem 3.2.7. There, we show that a tuple (X1, . . . , Xm) ∈Matp,q is in the nullcone of the left-right action if and only if there exist subspacesU ⊆ Kq, V ⊆ Kp with the property that

q dimU > p dimV

Thus, we can see King’s stability condition as a generalization of Theorem 3.2.7 toarbitrary quivers.

4.3.2 Schofield Semi-Invariants

In this section, we assume that Q is a quiver with no oriented cycles.

Recall that given representations V ∈ Rep(Q,α) and W ∈ Rep(Q,β), the map dVW isdefined as

dVW :"

x∈Q0

HomK(V (x),W (x)) →"

a∈Q1

HomK(V (ta),W (ha))

(φ(x) | x ∈ Q0) )→ (φ(ha)V (a))−W (a)φ(ta) | a ∈ Q1)

In terms of coordinates, dVW is the |Q1|× |Q0| block matrix

dVW =

%

&&&&'

Ba1,x1 Ba1,x2 . . . Ba1,xr

Ba2,x1

. . ....

.... . .

...

Bas,x1 . . . . . . Bas,xr

(

))))*

where x1, . . . , xr are the vertices, a1, . . . , as are the arrows and Ba,x is the α(ta)β(ha)×α(x)β(x)-matrix

Ba,x =

>????@

????A

0 if ha ∕= x ∕= ta

−Iα(x) ⊗W (a) if ha ∕= x = ta

V (a)T ⊗ Iβ(x) if ha = x ∕= ta

V (a)T ⊗ Iβ(x) − Iα(x) ⊗W (a) if ha = x = ta

Here Ik denotes the k × k-identity matrix and ⊗ is the Kronecker product of matrices.We give the following example for an explanation of the above construction.

Example 4.3.5. Let n ∈ Z≥0 and let V = (X1, . . . , Xm),W = (Y1, . . . , Ym) ∈ Matmn,n. Wecan consider V and W as two representations of the generalized Kronecker quiver withm arrows with dimension vector (n, n). Then, the map dVW is given by

dVW (A,B) = (BX1 − Y1A,BX2 − Y2A, . . . , BXm − YmA)

where A,B ∈ Matn,n.



Given a matrix N ∈ Matp,q, let v(N) denote the pq × 1-column vector obtained byappending the columns of N into a single column. Then, it is well-known that for amatrix M ∈ Matr,p we have

v(MN) = (Iq ⊗M)v(N).

Similarly, for a matrix M ∈ Mat q, r we have

v(NM) = (MT ⊗ Ip)v(N).

In other words, if LM denotes the linear map

LM : Matp,q → Matp,q

N )→ MN,

then the matrix of LM equals Iq ⊗M . Similarly, if RM is the right multiplication map,then the matrix of RM is MT ⊗ Ip.

In particular, we write

dVW (A,B) =

%

&'−In ⊗ Y1 XT

1 ⊗ In...

...−In ⊗ Ym XT

m ⊗ In

(

)*7AB

8

The construction of the matrix of dVW for different quivers is similar.

The action of GLα×GLβ on dVW is as follows: Given g ∈ GLα, h ∈ GLβ , set V′ = g ·V

and W ′ = h ·W . ThendV

′W ′ = A · dVW · C

where A is the |Q1|× |Q1|-block diagonal matrix

A =

%

&&&'

(g(ta1)−1)T ⊗ h(ha1)

(g(ta2)−1)T ⊗ h(ha2)

. . .

(g(tas)−1)T ⊗ h(has)

(

)))*

and C is the |Q0|× |Q0|-block diagonal matrix

C =

%

&&&'

g(x1)T ⊗ h(x1)

−1

g(x2)T ⊗ h(x2)

−1

. . .

g(xr)T ⊗ h(xr)

−1

(

)))*

We have

det(A) =

Ba∈Q1

det(h(ha))α(ta)B

a∈Q1det(g(ta))β(ha)

, det(C) =

Bx∈Q0

det(g(x))β(x)B

x∈Q0det(h(x))α(x)

.



Given a dimension vector α, we define the group character σ = 〈α, ·〉 to be

σ(x) = α(x)−+

a∈Q1,ha=x

α(ta)

Observe that if we take g to be the identity element g = (idV (x) | x ∈ Q0) ∈ GLα, wehave

det(A) det(C) = σ−1(h) = σ(h−1). (4.2)

Similarly, for a dimension vector β we define the group character τ = 〈·,β〉 to be

τ(x) = β(x)−+

a∈Q1,ta=x

β(ha)

Once again, for h = (idW (x) | x ∈ Q0) ∈ GLβ we have

det(A) det(C) = τ(g).

Definition 4.3.6. Assume that V ∈ Rep(Q,α) and β is a dimension vector suchthat 〈α,β〉 = 0 so dVW is a square matrix. We define the Schofield semi-invariantcV : Rep(Q,β) → K to be

cV (W ) = det dVW .

Here, det is taken with respect to an arbitrary choice of bases for V and W , so cV isdefined up to a scalar.

Similarly, assume thatW ∈ Rep(Q,β) and α is a dimension vector such that 〈α,β〉 = 0.We define the Schofield semi-invariant cW : Rep(Q,α) → K to be

cW (V ) = det dVW .

Proposition 4.3.7. Let α,β be two dimension vectors such that 〈α,β〉 = 0. GivenV ∈ Rep(Q,α), the Schofield semi-invariant cV is a quiver semi-invariant of weight〈α, ·〉. Similarly, given W ∈ Rep(Q,β) we have cW ∈ SI(Q,α)−〈·,β〉.

Proof. Let W ∈ Rep(Q,β) and h ∈ SLβ . Set σ = 〈α, ·〉. Then

(h · cV )(W ) = cV (h−1 ·W )

= det dVh−1·W

(use (4.2)) = σ−1(h−1) det dVW

= σ(h) det dVW

= σ(h)cV (W )

so h · cV = σ(h)cV which implies that cV ∈ SI(Q,β)〈α,·〉. The proof of the second partis similar.



Remark 4.3.8. Let Q be the generalized Kronecker quiver θ(m). We know the semi-invariants of Q from Section 3.2.3. We want to show that the semi-invariants of theform det(t1X1 + · · ·+ tmXm) and the Schofield semi-invariants of weight (1,−1) are inone-to-one correspondence.

Label the vertices of Q0 as 1 and 2 such that each arrow has tail 1 and head 2. LetV be the Q-representation with dimension vector dimV = (1,m− 1) and on each arrowai, i = 1, . . . ,m we set

V (ai) =

;0 if i = 1

ei−1 otherwise

Here, with a slight abuse of notation, ei denotes the map K → Km−1 mapping 1 to thei-th standard vector ei.

Let β = (n, n). Then

〈α,β〉 =+

x∈Q0

α(x)β(x)−+

a∈Q1

α(ta)β(ha)

= mn−mn = 0

Thus, by the previous proposition, cV is a semi-invariant of Rep(Q,β) of weight σ =〈α, ·〉 = (1,−1). In fact, for W ∈ Rep(Q, (n, n)) we have

cV (W ) = det(−W (a1))

which has weight (1,−1) as

(g(1), g(2)) · (W (a1), . . . ,W (am)) = (g(2)W (a1)g(1)−1, . . . , g(2)W (am)g(1)−1).

We leave the proof of the following fact to the reader: If f1, . . . , fm : K → Km−1 spanHom(K,Km−1) then for V ′ ∈ Rep(Q,α) with V ′(ai) = fi we have

cV′(W ) = det(t1W (a1) + · · ·+ tmW (am))

where up to a sign, each ti is an m− 1×m− 1-minor of the matrix%

&&&'

fT1

fT2...

fTm

(

)))*

Proposition 4.3.9. Let α,β be dimension vectors such that 〈α,β〉 = 0.

1. Let V ∈ Rep(Q,α) and consider the semi-invariant cV ∈ SI(Q,β)〈α,·〉. Supposethat we have an exact sequence of Q-representations

0 → V1 → V → V2 → 0

where dim(Vi) = αi for i = 1, 2. If 〈α1,β〉 > 0, then cV = 0. If 〈α1,β〉 = 0 thenwe have cV = cV1cV2.



2. Let W ∈ Rep(Q,β) and consider the semi-invariant cW ∈ SI(Q,α)−〈·,β〉. Supposethat we have an exact sequence of Q-representations

0 → W1 → W → W2 → 0

where dim(Wi) = βi for i = 1, 2. If 〈α,β1〉 < 0, then cW = 0. If 〈α,β1〉 = 0 thenwe have cW = cW1cW2 .

Proof. We will prove the first part of the theorem as the second part is similar. For eachx ∈ Q0, let

ex1 , ex2 , . . . , e

xα1(x)

be a basis of V1(x) and extend each such basis to a basis

ex1 , ex2 , . . . , e

xα(x)

of V (x). Consider the space spanned by exα1(x)+1, . . . , exα(x) for each x ∈ Q0. Then this

space is isomorphic to V2(x) and we have an isomorphism V (x) = V1(x) ⊕ V2(x). Asimilar decomposition also holds for the linear maps if we define

X1 ="

x∈Q0

Hom(V1(x),W (x)), X2 ="

x∈Q0

Hom(V2(x),W (x))

andY1 =

"

a∈Q1

Hom(V1(ta),W (ha)), Y2 ="

a∈Q1

Hom(V2(ta),W (ha)).

Then we can write dVW as a map

dVW : X1 ⊕X2 → Y1 ⊕ Y2.

Now, dVW maps X1 into Y1: If f = (f(x) | x ∈ Q0) ∈ X1, then

dVW (f) = (f(ha)V (a)−W (a)f(ta) | a ∈ Q1)

and since V (a) maps V1(ta) to V1(ha) (V1 is a sub-representation), we see that dVW (f) ∈Y1.

By the definition of the inner product, we have

dimX1 − dimY1 = 〈α1,β〉.

If this number is greater than 0, then dVW is not injective so cV = 0. If this numberequals 0, then since dVW maps X1 to Y1, we can write

dVW =

7dV1W ∗0 dV2

W

8

where the blocks on the diagonal are square matrices. Thus, we have

cV (W ) = det dVW = det dV1W det dV2

W = cV1(W )cV2(W )

and we obtain the desired result.



Proposition 4.3.10. Let Q be a quiver without oriented cycles and let β be a dimensionvector. If SI(Q,β)σ ∕= 0, then there exists a dimension vector α such that σ = 〈α, ·〉.

Proof. Without loss of generality, assume that Q0 = 1, . . . , n and each arrow a ∈ Q1

satisfies ha > ta. We can assume this since Q has no oriented cycles.

Let α be the dimension vector given by

α(i) = σ(i) ++

j∈Q0

p(j, i)σ(y)

where p(j, i) denotes the number of distinct paths from j to i. Observe that p(j, i) isnecessarily finite as Q has no oriented cycles.

We want to prove that σ = 〈α, ·〉. Given vertices i, j, let r(i, j) denote the number ofarrows from i to j. Then we have

(〈α, ·〉)(k) = α(k)−+

a∈Q1,ha=k

α(ta)

= σ(k) ++

i<k

p(i, k)σ(i)−+

a∈Q1,ha=k

#σ(ta) +

+

j<ta

p(j, ta)σ(ta)$

= σ(k) ++

i<k

p(i, k)σ(i)−+

i<k

r(i, k)σ(i)−+

i<k

+

j<i

r(i, k)p(j, i)σ(j)

= σ(k)

The last equality holds as given a vertex i < k, each path from i to k is either an arrowfrom i to k or it is the concatenation of an arrow from i to k with a path from j to i forsome j, i.e.

+

i<k

p(i, k)σ(i) =+

i<k

r(i, k)σ(i)−+

i<k

+

j<i

r(i, k)p(j, i)σ(j)

Now we can state the main theorem of this section which states that the ring ofsemi-invariants is (linearly) spanned by Schofield semi-invariants:

Theorem 4.3.11 ([DW00],[SV99]). Let Q be a quiver without oriented cycles and letβ be a dimension vector. The space of semi-invariants of Rep(Q,β) of weight 〈α, ·〉 islinearly spanned by Schofield semi-invariants cV where V ∈ Rep(Q,α).

Proof. There will be three steps. In the first step we enlarge Q to a new quiver Pby adding two more vertices. Moreover, we will introduce a dimension vector γ ofP satisfying γ |Q= β and a new weight τ of P such that τ(x) = 0 for all x ∈ Q0.



In the second step, we show that for every quiver Q and weight σ, we can deletevertices of weight 0 from Q in a special way such that the resulting quiver will have anisomorphic space of semi-invariants with the original one.

When the above process is applied to every vertex of P of weight 0, as τ is 0 on Q, theresulting quiver will have only two vertices. Moreover, on both vertices we will have thesame dimension and the weight will be (1,−1). This quiver is the generalized Kroneckerquiver and then using Remark 4.3.8 and Theorem 3.2.26 we will show that the theoremholds for this case.

Step 1: Construct a new quiver P as follows:

Let P0 = Q0∪s+, s− be the vertex set and P1 = Q1∪P+∪P− be the arrow set whereP+ contains maxσ(xi), 0 arrows from s+ to xi for i = 1, . . . , |Q0| and P− containsmax−σ(xi), 0 arrows from xi to s− for i = 1, . . . , |Q0|. For example, if Q is thegeneralized Kronecker quiver θ(m) with σ = (d,−d), then P is the quiver

s+ ... x ...y ...

s−

p1

pd

a1

am

q1

qd

Here, the middle portion is our original quiver Q.

Define a new weight τ such that τ(s+) = 1, τ(s−) = −1 and τ(x) = 0 for x ∈ Q0.Moreover, let γ be the dimension vector given by

γ(s+) =+

σ(x)>0

σ(x)β(x), γ(s−) =+

σ(x)<0

−σ(x)β(x), and γ(x) = β(x) for x ∈ Q0.

We claim that SI(P, γ)τ is isomorphic to SI(Q,β)σ.

Observe that given a representation V ∈ Rep(P, γ) and an arrow a : s+ → x ∈ P1,the corresponding matrix V (a) has size β(x)× γ(s+) = β(x)×

,σ(x)>0 σ(x)β(x). Since

there are σ(x) many arrows from s+ → x, the block matrix

%

&&&'

V (c1)

V (c2)...

V (ck)

(

)))*

is square, where ci are all the arrows emanating from s+. Denote the determinant ofthis matrix by det(+). Similarly, we form the block matrix

1V (d1) V (d2) . . . V (dk)

2

where di are all the arrows towards s− and denote its determinant by det(−).



We claim that the linear map

ψ : SI(Q,β)σ → SI(P, γ)τ

F )→ det(+)F det(−)

is an isomorphism. It is easy to see that ψ is injective as ker(ψ) = 0. The fact that ψis surjective follows by the First Fundamental Theorem of Invariant Theory for SLn (seeTheorem 2.2.24): Observe that the polynomials on Rep(P, γ) on which GLs+ × GLs−

acts with weight (1,−1) are all of the form det(+)F det(−) for some F ∈ K[Rep(Q,β)].We want to check for which F GLβ acts trivially on det(+)F det(−). Now, we claimthat GLβ acts on det(+) det(−) with weight −σ as for g ∈ GLβ we have

g·det(+) = det

3%

&&&'

g−1(t(c1))g−1(t(c2))

. . .

g−1(t(ck))

(

)))*

%

&&&'

V (c1)

V (c2)...

V (ck)

(

)))*

4=

!

x∈Q0,σ(x)>0

(det g(x))−σ(x)

and similarly

g · det(−) =!

x∈Q0,σ(x)<0

(det(g(x))−σ(x).

Thus we obtain

g · det(+) det(−) =!

x∈Q0

(det(g(x))−σ(x) det(+) det(−)

which implies that if det(+)F det(−) ∈ SI(P, γ)τ , then F must be of weight σ.

In the next step, we will show that the theorem holds for Rep(P, γ). However, beforeproceeding to next step, we still need to show that the inverse image of a Schofield semi-invariant in SI(P, γ)τ is a Schofield semi-invariant in SI(Q,β)σ. To this end, let δ bethe dimension vector of P satisfying τ = 〈δ, ·〉 and let V ∈ Rep(P, δ).

Claim : There exists W ∈ Rep(Q,β)σ such that ψ(cW ) = cV .

Proof : Without loss of generality we assume that cV ∕= 0 since otherwise we can takeW = 0.

Let σ+ be the weight of det(+) under the action of GLγ , i.e.

σ+(x) =

;0 if σ(x) ≤ 0

−σ(x) otherwise

for x ∈ Q0 and σ+(s+) = 1,σ+(s−) = 0. Similarly, let σ−(x) be the weight of det(−).Now, we claim that there exist dimension vectors γ+ and γ− of P satisfying σ+ = 〈γ+, ·〉and similarly σ− = 〈γ−, ·〉. Define

γ+(s+) = 1, γ+(s−) = 0, γ+(x) = 0 for x ∈ Q0.



Then we have

(〈γ+, ·〉)(s+) = γ+(s+)−+

a∈P1,ha=s+

γ+(ta) = γ+(s+) = 1 = σ+(s+)

(〈γ+, ·〉)(s−) = γ+(s−)−+

a∈P1,ha=s−

γ+(ta) = 0 = σ+(s−)

(〈γ+, ·〉)(x) = γ+(x)−+

a∈P1,ha=x

γ+(ta) = −2a ∈ P1 | ha = x, ta = s+ = σ+(x).

Thus, σ+ = 〈γ+, ·〉. Similarly, we define

γ−(s+) = 0

γ−(s−) = −1 ++

x∈Q0,σ(x)<0

+

y∈Q0,σ(y)<0

by,xσ(x)σ(y)

γ−(x) = −+

y∈Q0,σ(y)<0

by,xσ(y)

where bx,y denotes the number of distinct paths from x to y. Then, a direct computationshows that

σ− = 〈γ−, ·〉.

Moreover, we have

〈γ+, γ〉 =+

x∈P0

γ+(x)γ(x)−+

a∈P1

γ+(ta)γ(ha)

= γ+(s+)γ(s+)−+

x∈Q0,σ(x)>0

σ(x)γ+(s+)γ(x)

=+

x∈Q0,σ(x)>0

σ(x)β(x)−+

x∈Q0,σ(x)>0

σ(x)β(x) = 0.

Similarly we have 〈γ−, γ〉 = 0.

Consider the representation V1 = V |P−s+, i.e. the restriction of V to P − s+.Note that this representation satisfies

V1(s+) = 0, V1(a) = 0 if ta = s+

and it is equal to V for any other vertex or arrow. V1 is obviously a sub-representationof V as there is no arrow x → s+. Hence, there is an exact sequence

0 → V1 → V → V2 → 0

where V2 satisfies

dim(V2)(x) =

;0 if x ∕= s+

dim(V (s+)) = δ(s+) if x = s+.



On the other hand, as τ = 〈δ, ·〉, we have

δ(s+) = τ(s+) ++

x∈P0

bx,s+τ(x) = τ(s+) = 1.

Hence, dim(V2) = γ+.

Now, since 〈γ+, γ〉 = 0, by Proposition 4.3.9, we have

cV = cV1cV2 .

On the other hand, since cV2 is a semi-invariant of weight

〈γ+, ·〉 = σ+,

it must be a constant multiple of det(+). Without loss of generality, we may assumethat cV2 = det(+), as multiplying each linear map V (a) with a constant does not changethe result. We obtain

cV = cV1 det(+).

We have already extracted det(+). Now we need to extract det(−) by showing cV1 canbe written as cV1 = cV

′det(−) for some V ′.

Recall the definition of the representation Px from Definition 4.1.14: Px is the repre-sentation such that Px(y) is spanend by [x, y] where [x, y] is the set of all all distinct pathsfrom x to y and Px(a) is obtained by linearly extending f : [x, ta] → [x, ha] mappingp )→ ap.

Let M be the representation satisfying

0 → Ps− →"

ha=s−

Pta → M → 0.

Here, for each a ∈ Q1 with ha = s−, the maps Ps− → Pta is given as follows: Sincethere is no arrow s− → x, for each s− ∕= x ∈ P0 we have Ps−(x) = 0. The only non-trivial space is Ps−(s−) which is 1-dimensional and spanned by the trivial path froms− to s−, denoted by es− . We map Ps−(s−) to Pta(s−) = 〈[ta, s−]〉 by es− )→ a. It isstraightforward to check that this defines a morphism of P -representations.

Now, we have

dim(Ps−)(s−) = 1, dim(Ps−)(x) = 0 for s− ∕= x ∈ P0.

Also,

dim("

ha=s−

Pta)(x) =

>?@

?A

0 if x = s+

−,

y∈Q0,σ(y)<0 by,xσ(y) if x ∈ Q0,z∈Q0,σ(z)<0

,y∈Q0,σ(y)<0 by,zσ(z)σ(y) if x = s−



where bx,y denotes the number of distinct paths from x to y. This holds because in thecase x = s+, all spaces Pta(s+) = 〈[ta, s+]〉 are trivial since there are no arrows havings+ as head. The case x ∈ Q0 holds because

dim("

ha=s−

Pta)(x) =+

ha=s−

bta,x.

We can change this summation as follows: For each y ∈ Q0 with σ(y) < 0, there are−σ(y) arrows from y to s−. Thus, instead of counting the arrows, we can count thevertices satisfying σ(y) < 0 and write

+

ha=s−

bta,x = −+

y∈Q0,σ(y)<0

by,xσ(y).

Lastly, the formula hols for x = s− since we have

dim("

ha=s−

Pta)(s−) =+

ha=s−

2[ta, s−].

We can once again count this number by summing over all vertices y satisfying σ(y) < 0:For each y with σ(y) < 0, there are −σ(y) many arrows from y to s−. Also, the numberof paths from y to s− equals −

,σ(z)<0 by,zσ(z). Thus,we obtain

dim("

ha=s−

Pta)(s−) =+

ha=s−

bta,s− =+

σ(y)<0

+

σ(z)<0

by,zσ(y)σ(z).

Using the formulas above, we deduce that

dim(M) = dim("

ha=s−

Pta)− dim(Ps−) = γ−.

Thus, cM is a semi-invariant of weight σ− which implies that cM equals det(−) up toa non-zero scalar multiple. Once again, without loss of generality we may assume thatcM = det(−). Observe that

dimHomQ(M,V1)− dimExtQ(M,V1) = 〈γ−, δ − γ+〉 = σ− · (δ − γ+) = σ− · δ − σ− · γ+.

We have

σ−·δ =+

x∈Q0

σ−(x)δ(x)+σ−(s+)δ(s+)+σ−(s−)δ(s−) = −+

σ(x)<0

σ(x)bs+,x+0+(1−bs+,s−) = 1

and

σ− · γ+ =+

x∈Q0

σ−(x)γ+(x) + σ−(s+)γ+(s+) + σ−(s−)γ+(s−) = 0.



We deduce that dimHomQ(M,V1) − dimExtQ(M,V1) = 1 which implies that thereexists a non-zero morphism µ : M → V1. Set V3 to be the image of µ so V3 is asub-representation of V1.

We want to show that 〈dim(V3), γ〉 = 0 so cV3 exists. Since det(+)cV1 = cV ∕= 0, wehave cV1 ∕= 0. Moreover, cM = det(−) ∕= 0. Consider the exact sequences

0 → V3 → V1 → V4 → 0

and

0 → N → M → V3 → 0.

Here, N = ker(µ). Since cV1 ∕= 0 and cM ∕= 0, by Proposition 4.3.9, we have 〈dim(V3), γ〉 ≤0 and −〈dim(V3), γ〉 = 〈N, γ〉 ≤ 0. Thus, we have 〈dim(V3), γ〉 = 0. Using Proposition4.3.9, we deduce that det(−) = cM = cNcV3 . However, since det(−) is an irreduciblepolynomial, cV1 must be det(−) (it cannot be constant since µ ∕= 0).

We obtained cV = det(+)cV4 det(−). Let V5 = V4 |s− be the restriction of V4 to s−.There is an exact sequence

0 → V5 → V4 → V6 → 0.

Set W = V6 |Q. We claim that ψ(cW ) = cV .

For a representation U ∈ Rep(Q,β), we define the extension of U ′ to P as

U ′(s+) ="

a∈P1,ta=s+

U(ha), U ′(s−) ="

a∈P1,ha=s−

U(ta),

where for an arrow a with ta = s+, we define U ′(a) to be the canonical surjection ofU ′(s+) onto U(ha) and for an arrow a with ha = s−, we define U

′(a) to be the canonicalinjection of U(ta) into U ′(s−). Let U1 = U ′ |s− . Then there is an exact sequence

0 → U1 → U ′ → U2 → 0.

Let U3 = U2 |P−s+ , so there is another exact sequence

0 → U3 → U2 → U4 → 0.

Observe that U3 |Q is nothing but U . We have

cV (U ′) = cV4(U ′) (this holds since det(+)U(s+) = 1, det(−)U(s−) = 1)

= cV4(U1)cV4(U3)c

V4(U4)

= cV4(U3) (since cV4(U1) cV4(U4) are constants)

= cV5(U3)cV6(U3)

= cV6(U3) (since cV5(U3) is constant)

= cW (U).



Thus, ψ(cW ) = cV and the claim is proven.

Step 2: In this step, we show that we can construct a new quiver R by deleting avertex of weight 0 from P without changing the space of semi-invariants.

Given a vertex x ∈ P of weight 0, let a1, . . . , as be the arrows satisfying hai = x andlet b1, . . . , br be the set of arrows satisfying tbi = x. We define a new quiver R as follows:

Set R0 = P0 − x and

R1 = a ∈ P1 | ha ∕= x ∕= ta ∪ aibj : tai → hbj | i = 1, . . . , s, j = 1, . . . , r

In other words, for each arrow pair (a, b) satisfying ha = x = tb, we add an arrow fromta to hb. For example, if P is the quiver given as an example in the beginning of theproof, then R is the following quiver:

s+ ...y ...

s−

p11

pdm

a1

am

There are dm arrows from s+ to y as there are d arrows from s+ to x and m arrowsfrom x to y.

Let γ′ be the restriction of γ to R and τ ′ be the restriction of τ to R.

We define a map res : Rep(P, γ) → Rep(R, γ′) as follows:

For c ∈ P1 with hc ∕= x ∕= tc, set res(V )(a) = V (a). For i ∈ [s], j ∈ [r], defineres(V )(aibj) = V (bj)V (ai). We call this map the restriction map.

The action of the group GLγ(x) on Rep(P, γ) is given by

(g · V )(c) =

>?@

?A

V (c) if hc ∕= x ∕= tc

gV (ai) if c = ai

V (bj)g−1 if c = bj .

By the Fundamental Theorem of Invariant Theory for GLn (see Theorem 2.2.23 andthe following geometric interpretation), res is the categorical quotient map with respectto the action of GLγ(x). As a result, we have a surjection

res( : K[Rep(R, γ′)] → K[Rep(P, γ)]GLγ(x) .

Moreover, res is GLγ′-equivariant as for G ∈ GLγ′ we have

res(G · V ) = res#(G(hc)V (c)G(tc)−1 | hc ∕= x ∕= tc), (G(hai)V (ai) | i ∈ [s]), (V (bj)G(tbj)

−1 | j ∈ [r])$

= (G(ha) res(V )(a)G(ta)−1 | a ∈ R1)

= G · res(V ).



Thus, res( induces a surjection

res( : SI(Rep(R, γ′))τ ′ → SI(Rep(P, γ))τ .

Assume that the theorem holds for R. If we show that ψ( maps Schofield semi-invariants to Schofield semi-invariants, then the theorem also holds for P . We achievethis via a map

ind : Rep(R,α′) → Rep(P,α)

which is the left adjoint functor of res.

Let τ ′ = 〈α′, ·〉. Extend α′ to a dimension vector α of P via

α(y) =

;α′(y) if y ∈ R0,s

i=1 α′(tai) if y = x.

We claim that 〈α, ·〉 = τ . To prove this, there are three cases we consider : Firstly, wehave

(〈α, ·〉)(x) = α(x)−+

a∈P1,ha=x

α(ta) =

s+

i=1

α′(tai)−s+

i=1

α′(tai) = 0 = τ(x).

If y is a vertex such that there is no arrow x → y, then we have

(〈α, ·〉)(y) = α(y)−+

a∈P1,ha=y

α(ta) = α′(y)−+

a∈P1,ha=y

α′(ta) = τ ′(y) = τ(y).

Lastly, we prove the equality when there are arrows from x to y. Without loss ofgenerality, say hb1 = hb2 = · · · = hbk = y. Then we have

(〈α, ·〉)(y) = α(y)−+

a∈P1,ha=y

α(ta)

= α′(y)− kα(x)−+

a∈P1,ha=y,a ∕=bi

α′(ta)

= α′(y)− k

s+

i=1

α′(tai)−+

a∈P1,ha=y,a ∕=bi

α′(ta)

= α′(y)−+

a∈R1,ha=y

α′(ta) = τ ′(y) = τ(y).

Here, the equality holds as if hbi = y then there are s arrows tai → y in the quiver R.

Now we define ind : Rep(R,α′) → Rep(P,α) as

ind(V )(y) =

;V (y) if y ∕= x<s

i=1 V (tai) if y = x



on vertices and

ind(V )(c) =

>?@

?A

V (c) if c ∕= ai, bj

ık if c = ak,si=1 V (aibk)πi if c = bk

where ik denotes the canonical injection of V (tak) into<s

i=1 V (tai) = ind(V )(x) and πjdenotes the canonical projection of ind(V )(x) =

<si=1 V (tai) onto V (taj).

Step 3: In the first step, we constructed a quiver P from Q by adding two morevertices s+, s−. Then we defined a new dimension vector γ such that γ(s+) = γ(s−).Moreover, we defined a new weight τ on P such that τ(s+) = 1, τ(s−) = −1 and τ is 0on Q0.

In the second step, we have shown that we can delete vertices of weight 0 withoutchanging the ring of semi-invariants. Using this process for each vertex of Q, we end upin a quiver with two vertices, each one has the same dimension and the weight is (1,−1).

Thus, we reduced the theorem to the case where Q is the generalized Kronecker quiver.Recall that in Theorem 3.2.26, we show that the semi-invariants of the generalizedKronecker quiver of weight (1,−1) are spanned by polynomials of the form det(t1X1 +· · ·+ tmXm) where Xi are the linear maps assigned to each arrow ai of Q. Moreover, inRemark 4.3.8, we show that the semi-invariants of this form and Schofield semi-invariantsare in one-to-one correspondence. Thus, the theorem holds for the generalized Kroneckerquiver and by the previous steps it holds in generality.

The above theorem describes a generating set for the ring of semi-invariants of quiverrepresentations. There is an alternative way to describe a set of generators as determi-nants of block matrices.

Theorem 4.3.12 ([DW00]). Let Q be a quiver without oriented cycles, let α be a di-mension vector and σ be a weight satisfying α · σ = 0. Set σ+(x) = maxσ(x), 0 andσ−(x) = max−σ(x), 0. For each representation V ∈ Rep(Q,α) we define a map

A :"

x∈Q0

V (x)σ+(x) →"

x∈Q0

V (x)σ−(x)

where each block in Hom(V (x), V (y)) is given by tx,y1 V (p1)+ · · ·+ tx,yr V (pr) where pi areall paths from x to y and tx,y1 , . . . , tx,yr are indeterminates. Then SI(Q,α)σ is spannedby the coefficients of det(A) seen as a polynomial in variables tx,yi . In other words wehave

SI(Q,α)σ = 〈det(A(tx,yi = λx,yi )) | x, y ∈ Q0,λ

x,yi ∈ K〉.

Proof. See [DW00], Corollary 3.

Example 4.3.13. Consider the Kronecker quiver θ(m) with two vertices x, y andm arrowsa1, . . . , am from x to y. We stated before that the semi-invariants of θ(m) with dimension



vector (n, n) are matrix semi-invariants which are spanned by the polynomials of the form

det(T1 ⊗X1 + · · ·+ Tm ⊗Xm), T1, . . . , Tm ∈ Matmd,d .

The same result can be obtained using Theorem 4.3.12 above: For the dimension vector(n, n), we pick a weight σ = (d,−d) for d ∈ Z. If d ≥ 0, then we have σ+ = (d, 0) andσ− = (0, d) so "

v∈Q0

V (v)σ+(v) = V (x)d,"

v∈Q0

V (v)σ−(v) = V (y)d

which implies that A is the d× d block matrix

A =

%

&&&&'

t1,11 V (a1) + · · ·+ t1,1m V (am) . . . . . . t1,d1 V (a1) + · · ·+ t1,dm V (am)...

. . ....

.... . .

...

td,11 V (a1) + · · ·+ td,1m V (am) . . . . . . td,d1 V (a1) + · · ·+ td,dm V (am)

(

))))*.

Observe that A = T1 ⊗X1 + · · ·+ Tm ⊗Xm for Ti = [tj,ki ].

If d < 0, we have

"

v∈Q0

V (v)σ+(v) = V (y)d,"

v∈Q0

V (v)σ−(v) = V (x)d

but as there are no paths from y to x, we do not have any semi-invariants of weight(d,−d) for d < 0.

4.4 Degree Bounds

In this section, we give degree bounds for the ring of invariants and semi-invariants ofquivers. We use the terminology from Section 3.3: β denotes the minimum degree forwhich the invariants of degree at most β generate the ring of invariants and we use γ forthe bound of the degrees of invariants that cut out the nullcone.

Recall from the Section 3.3 that in characteristic 0, the ring of matrix invariants isgenerated by polynomials of degree at most n2. Le Bruyn and Procesi generalized toarbitrary quivers in [LP90] and proved that the invariants of degree at most N2 spanthe ring of invariants where N =

,x∈Q0

α(x) and α is the dimension vector. We give analternative proof to this theorem. Recall that in Corollary 4.2.5, we showed that thereexists a surjective, degree 1 ring map

ψ( : S(N,M) → K[Rep(Q,α)]GLα

where M = |Q1|.



Theorem 4.4.1 ([LP90]). Assume that char(K) = 0. Then the invariant ring K[Rep(Q,α)]GLα

is generated by invariants of degree ≤ N2 where N =,

i αi.

Proof. Since there is a surjective degree 1 ring map

ψ( : S(N,M) → K[Rep(Q,α)]GLα ,

we have β(K[Rep(Q,α)]GLα) ≤ β(S(N,M)) ≤ N2 and the result follows.

Similarly, we use the degree bound for S(N,M) in positive characteristic:

Theorem 4.4.2 ([DM16b]). We have β(K[Rep(Q,α)]GLα) ≤ (M + 1)N4 where M =|Q1| and N =

,x∈Q0

α(x).

Proof. Use the same surjective ring map and Theorem 3.3.17.

Now we focus on bounds for semi-invariants of quivers. Recall that the ring of semi-invariants has a weight decomposition

SI(Q,α) ="

σ

SI(Q,α,σ).

It is not easy to give bounds on a generating set of invariant in terms of the number ofvertices or edges. Instead, the bounds we give in the rest of the section will be in termsof σ.

Assume that Q has no oriented cycle. In the previous section, we show that givenσ ∈ Zn, if σ · α ∕= 0 then there is no non-zero semi-invariant of weight σ. Given σ withσ ·α = 0, we set σ = σ++σ− where σ+(x) = maxσ(x), 0 and σ−(x) = max−σ(x), 0.Then the invariants of the form det(A) for

A :"

x∈Q0

V (x)σ+(x) →"

x∈Q0

V (x)σ−(x)

generate the ring of semi-invariants. Here, A is the block matrix described in Theorem4.3.12.

Given a representation V ∈ Rep(Q,α) we define

C(V ) = σ | V is σ-semi-stable ⊆ ZQ0 .

By the King’s stability condition (Theorem 4.3.1), C(V ) is defined by the linear equationσ ·α = 0 and by linear inequalities σ ·dim(W ) ≤ 0 for sub-representations W of V . Thus,C(V ) is a convex polyhedra. Let L ⊆ C(V ) be an extremal ray of C(V ), i.e. assume thatQL is a 1-dimensional subspace of QQ0 and the half-line Q≥0L ⊆ QQ0 is contained in theboundary of Q≥0C(V ). As QL is in the boundary, it is defined by degenerating some of



the inequalities σ · dim(W ) ≤ 0 to equalities, i.e. there exist some sub-representationsW1, . . . ,Wk such that the equalities

σ · α = 0 and ∀i,σ · dim(Wi) = 0

define QL.

Let σ be a minimal integer vector in L, so L = dσ | d ∈ Z≥0. As σ ∈ C(V ), V isσ-semi-stable so as in Remark 4.3.2, we can form a filtration

0 = V0 # V1 # · · · # Vk = V

of sub-representations of V such that each Vi+1/Vi is σ-stable. The following lemmashows that the equalities σ · αi = 0 where αi = dim(Vi+1/Vi) determine QL:

Lemma 4.4.3. We have

QL = τ ∈ QQ0 | ∀i, τ · αi = 0

where αi = dim(Vi+1/Vi).

Proof. As each Vi+1/Vi is σ-stable, σ satisfies σ · αi = 0 for all i. Thus σ ∈ τ ∈ QQ0 |∀i, τ ·αi = 0 and since σ ∈ L, we obtain QL ⊆ τ ∈ QQ0 | ∀i, τ ·αi = 0. For the reverseinclusion, recall that QL is defined by σ · dim(Wi) = 0 for some sub-representationsW1, . . . ,Wk of V . For each i, consider a filtration

0 = U0 # U1 # · · · # Us = Wi

such that each factor Uj+1/Uj is σ-stable. As Wi are sub-representations of V , each suchfactor is isomorphic to Vl+1/Vl for some l. As

dim(Wi) =

s+

i=1

dim(Ui/Ui−1),

the equation dim(Wi) = 0 is a consequence of the equations σ · αi = 0. Thus, theequations σ · αi = 0 define QL.

Definition 4.4.4. For α ∈ Qn, define

‖α‖1 =n+

i=1

|α|i, ‖α‖2 = (

n+

i=1

α2i )

1/2.

Theorem 4.4.5 ([DM16a]). Let V ∈ Rep(Q,α) such that C(V ) is non-empty. Thenthere exists σ ∈ C(V ) such that each coordinate σi of σ satisfies

|σi| ≤# ‖α‖1n− 1

$n−1.



Proof. Let L be an extremal ray of C(V ) and let σ be a minimal integer vector in L.Assume that

0 = V0 # V1 # · · · # Vk = V

is a filtration of V such that each Vi+1/Vi is σ-stable. Set αi = dim(Vi+1/Vi). By theabove lemma, the equations σ ·αi = 0 define QL. As QL is a 1-dimensional space, thereexist β1, . . . ,βn−1 ∈ α0,α2, . . . ,αk−1 such that the equations σ ·βi = 0, i = 1, . . . , n−1define QL. Let M be the (n−1)×n-matrix having βi as its i-th row. Since β1, . . . ,βn−1

define QL, they are linearly independent, so M has rank (n − 1). Moreover, note thatker(M) ⊆ Qn equals QL.

Now let N(i) be the (n−1)×(n−1)-minor of M obtained by deleting the i-th column.Set ui = (−1)iN(i). Then we have

βi · (u1, . . . , un) = det(

7M

βi

8) = 0

which implies that

M ·

%

&&&'

u1u2...un

(

)))*= 0.

Moreover, we have

|ui| = |N(i)|≤ ‖β1‖1 · ‖β2‖1 . . . ‖βn−1‖1

≤#‖β1‖1 + · · ·+ ‖βn−1‖1

n− 1

$n−1

=#‖β1 + · · ·+ βn−1‖1

n− 1

$n−1

Here, the second inequality follows by AM-GM inequality. The first inequality holdsbecause N(i) is an (n − 1) × (n − 1) of M , so it is a sum of monomials of the form±(β1)i1 · (β2)i2 . . . (βn−1)in−1 . Thus, its absolute value is bounded by the sum of theabsolute values of all monomials. This sum equals ‖β1‖1 · ‖β2‖1 . . . ‖βn−1‖1. The lastequality follows by direct computation (note that βi ∈ Zn

≥0).

As,k−1

i=0 αi = α and βi ∈ α0, . . . ,αk−1, we deduce that each entry of β1+ · · ·+βn−1

is less than or equal to the corresponding entry of α. Thus, we further see that

|ui| ≤# ‖α‖1n− 1

$n−1.

However, as ker(M) = QL, σ is a minimal integer vector in L and u = (u1, . . . , un) isintegral, u is an integer multiple of σ so σ satisfies

|σi| ≤ |ui| ≤# ‖α‖1n− 1

$n−1



and the result follows.

For a weight σ, we define |σ|α = σ+ · α. The above theorem also gives a bound for|σ|α :

Corollary 4.4.6 ([DM16a]). Let V ∈ Rep(Q,α) such that C(V ) is non-empty. Thenthere exists σ ∈ C(V ) such that

|σ|αα ≤# ‖α‖1n− 1

$n−1#‖α‖12

$=

‖α‖12(n− 1)n−1

.

Proof. As σ · α = 0, we have σ+ · α = σ− · α. Thus,n+

i=1

|σi|αi = σ+ · α+ σ− · α = 2|σ|α.

On the other hand, since |σi| ≤#‖α‖1n−1

$n−1, we obtain

2|σ|α =

n+

i=1

|σi|αi ≤# ‖α‖1n− 1

$n−1· ‖α‖1.

The result follows.

Lemma 4.4.7. Let V ∈ Rep(Q,α) and let 0 ∕= σ ∈ C(V ). Then for all d ≥ |σ|α − 1,there exists a semi-invariant f of weight dσ satisfying f(V ) ∕= 0.

Proof. Recall from Theorem 4.3.12 that the ring of semi-invariants of Rep(Q,α) of weightσ is linearly spanned by the determinant of the matrices of the form

A :"

x∈Q0

V (x)σ+(x) →"

x∈Q0

V (x)σ−(x)

where each block in Hom(V (x), V (y)) is given by tx,y1 V (p1)+ · · ·+ tx,yr V (pr) where pi areall paths from x to y and tx,y1 , . . . , tx,yr are indeterminates. Note that A can be writtenas a linear combination A = t1X1 + · · ·+ tmXm where

m =+

x∈Q0

+

y∈Q0

σ+(x)bx,yσ−(y)

where bx,y is the number of paths from x to y and each Xi is a |σ|α× |σ|α-matrix. Thus,there is a surjective ring homomorphism

ψ : R(|σ|α,m) →"

d

SI(Q,α)dσ

that maps det(X1⊗T1+ · · ·+Xm⊗Tm) to the polynomial that maps V to det(V (p1)⊗T1 + · · ·+ V (pm)⊗ Tm). Observe that ψ maps the invariants of degree dn to invariantsof weight dσ. In particular, as γ(R(|σ|α,m)) ≤ |σ|α − 1, for each d ≥ |σ|α − 1 we canfind an invariant f of weight dσ satisfying f(V ) ∕= 0.



Theorem 4.4.8 ([DM16a]). The nullcone of the action of SLα on Rep(Q,α) is cut outby invariants f of weight σ for

|σ|α ≤ ‖α‖2n14(n− 1)2n−2

.

Proof. Assume that V is a representation that is not in the nullcone. Then there exists0 ∕= σ ∈ C(V ) such that

|σ|α ≤ ‖α‖12(n− 1)n−1

.

By the above lemma, there exists an invariant f of weight (|σ|α)2 ≤ ‖α‖2n14(n−1)2n−2 . Thus, if

a representation V is in the zero set of all semi-invariants of weight at most‖α‖2n1

4(n−1)2n−2 ,

then V must be in the nullcone.

Lemma 4.4.9 ([DM16a]). Let f be a semi-invariant of weight σ. Then the homogeneouscomponents of f are non-trivial only for degrees between |σ|α and (n− 1)|σ|α.

Proof. By Theorem 4.3.12, the semi-invariants of weight σ is linearly spanned by deter-minants of |σ|α× |σ|α block matrices in which each block is a linear combination of pathsin Q. As Q has no oriented cycles, the number of edges of each such path is bounded byn− 1, so the entries of each block has degree bounded by n− 1. We obtain the desiredresult.

Remark 4.4.10. In [DM16a], the above lemma is stated for n instead of n − 1 and theproof is the same. However, since there are no oriented cycles, a path cannot have lengthn, so the bound can be taken to be n− 1.

Now we can state a theorem for γ(SI(Q,α)).

Theorem 4.4.11 ([DM16a]). The nullcone of the action of SLα on Rep(Q,α) is cut

out by polynomials of degree at most‖α‖2n1

4(n−1)2n−3 , i.e.

γ(SI(Q,α)) ≤ ‖α‖2n14(n− 1)2n−3

.

Proof. The proof follows by the above lemma together with Theorem 4.4.8.


5 Order Three Tensors

In this chapter, we turn our attention to order three tensors. Our aim in this chapteris to study the action of SLa × SLb × SLc on Ca ⊗ Cb ⊗ Cc and describe its nullcone,if possible. Our main tool in this chapter is a special equivariant map ϕ from orderthree tensors of a special format to forms that we will introduce in the second section.In the third section, we study the invariant theory of forms and state the well-knownresults of this subject. Using the equivariant map ϕ together with the given knowledgeof the invariant theory of forms, we give a characterization of the nullcone of the spaceC2 ⊗Cn ⊗Cn of two slices tensors for any n. The main result is that a two slices tensorT is in the nullcone if and only if the binary form ϕ(T ) is in the nullcone. We alsodescribe the nullcone of the space C3 ⊗ C3 ⊗ C3 of cuboids. In this case, in addition tothe condition that the ternary cubic ϕ(T ) is in the nullcone, we also need the assumptionthat the slices of the cuboid in each direction has a rank 1 section.

5.1 Preliminaries

In this section, we give a setup for the rest of the chapter. For the main definitions andtheorems, see Section 2.1.

An order three tensor is an element of U ⊗ V ⊗ W for some vector spaces U, V andW . There is a natural action of the group GL(U) × GL(V ) × GL(W ) on the spaceU ⊗ V ⊗W . On rank 1 tensors, the action is given via

(g, h, k) · u⊗ v ⊗ w = (gu)⊗ (hv)⊗ (kw).

Observe that there are no non-constant invariants of this action: Taking g(t) = t · idU ,we have

limt→0

(g(t), idV , idW ) · T = 0

for all T ∈ U ⊗V ⊗W . However, when we consider the action of the subgroup SL(U)×SL(V ) × SL(W ) on U ⊗ V ⊗ W , there are non-constant invariants. This action isthe ”three-dimensional analog” of the left-right action from Section 3.2. However, thetheory is much more complex than the left-right action. For example, in Section 3.2.1,we describe the nullcone of the left-right action using max-flow-min-cut theorem (orin the case of square matrices, Hall’s marriage theorem). This theorem has no threedimensional analog so it is much harder to describe the nullcone of the order threetensors.


Recall that the space U⊗V ⊗W is isomorphic as a GL(U)×GL(V )×GL(W )-moduleto the space Hom(U∗, V ⊗ W ). Moreover, as V ⊗ W ∼= Hom(W ∗, V ), we can identifyU ⊗ V ⊗ W with the space Hom(U∗,Hom(W ∗, V )). The action of GL(U) × GL(V ) ×GL(W ) on the latter space is given as

((g, h, k) · T )(ϕ) = h · T (g−1ϕ) · k−1, ϕ ∈ U∗.

We can also write this action in coordinates as follows : Assume that dimU =a, dimV = b, dimW = c and let (x1, . . . , xa) be a basis of U , (y1, . . . , yb) be a basis ofV and (z1, . . . , zc) be a basis of W . Denote the corresponding dual bases with xi, yj , zk

respectively. Then, each tensor T ∈ U ⊗ V ⊗W can be identified with a 3-dimensionalarray [T ijk] where T ijk ∈ C is the coefficient of xi ⊗ yj ⊗ zk in T , i.e.

T =+

i∈[a],j∈[b],k∈[c]T ijkxi ⊗ yj ⊗ zk.

The identification of the tensor product U ⊗ V ⊗ W with Hom(U∗, V ⊗ W ) is givenby

T : U∗ → V ⊗W

λ1x1 + · · ·+ λax

a )→+

i,j,k

λiTijkyj ⊗ zk.

Moreover, identifying V ⊗W = Hom(W ∗, V ), we write T as a map

T : U∗ → Hom(W ∗, V )

λ1x1 + · · ·+ λax

a )→ (zk )→+

i,j,k

λiTijkyj).

In coordinates, this identifications allow us to represent T with a tuple (X1, . . . , Xa) ofmatrices in Matb,c where Xi = T (xi). It is straightforward to see that for all i = 1, . . . , awe have

Xi =

%

&&&&'

T i11 T i12 . . . T i1c

T i21 . . ....

.... . .

...T ib1 . . . . . . T ibc

(

))))*.

Now the action of GLa × GLb × GLc on order three tensors translate to this tuplerepresentation as

(g, h, k) · (X1, . . . , Xa) = h#g11X1 + · · ·+ g1aXa, . . . , ga1X1 + · · ·+ gaaXa

$k−1. (5.1)

where gij denotes the (i, j)-th entry of the matrix g.



We also represent an order three tensor T with a single matrix MT as follows: Wepick formal variables z1, . . . , za and write

MT (z1, . . . , za) = z1X1 + · · ·+ zaXa

which is a matrix where each entry is a linear form in variables z1, . . . , za.

Example 5.1.1. Let

T = e1 ⊗ e1 ⊗ e1 + e2 ⊗ e2 ⊗ e2 ∈ C2 ⊗ C2 ⊗ C2

where ei denotes the i-th standard basis vector of C2.

Then T is identified with

T : (C2)∗ → Hom((C2)∗,C2)

e1 )→ (e1 )→ e1, e2 )→ 0)

e2 )→ (e1 )→ 0, e2 )→ e2)

so

T =#7

1 00 0

8,

70 00 1

8$

and

MT (z1, z2) =

7z1 00 z2

8.

Proposition 5.1.2. Let (g, h, k) ∈ GLa ×GLb ×GLc be given. Then we have

M (g,h,k)·T (z1, . . . , za) = hMT (gt · z)k−1

where gt · z denotes the vector

gt · z = (g11z1 + · · ·+ ga1za, . . . , g1az1 + · · ·+ gaaza).

Proof. This directly follows by Equation 5.1 above.

5.2 An Equivariant Map from Tensors to Forms

Now we consider the action of SLa × SLb × SLc on Ca ⊗ Cb ⊗ Cc. Given a tensor T ,let MT (z1, . . . , za) denote the matrix we defined in the previous section. Given a vector(c1, . . . , ca) ∈ Ca, we set

MT (c1, . . . , ca) = c1X1 + · · ·+ caXa ∈ Matb,c(C)

where Xi is the i-th slice of T in the first direction.

If f ∈ C[Matb,c] is a polynomial, then the evaluation of f at MT (z1, . . . , za) is definedto be the polynomial F ∈ C[z1, . . . , za] given by

F (c1, . . . , ca) = f(MT (c1, . . . , ca)).



Example 5.2.1. Let

MT (z1, z2) =

7z1 00 z2

8

Then for c1, c2 ∈ C we have

MT (c1, c2) =

7c1 00 c2

8

Consider the determinant polynomial det ∈ C[Mat2,2]. Then

det(MT (z1, z2)) = z1z2.

We identify C[z1, . . . , za] with the symmetric algebra S((Ca)∗). Then the action ofGLa on C[z1, . . . , za] is given by

(gt · f)(z1, . . . , za) = f(gt ·

%

&&&'

z1z2...za

(

)))*).

In particular, when b = c = n, we consider det ∈ C[Matn,n] and obtain the followingequivariant map from the vector space of order three tensors to the vector space offorms:

Proposition 5.2.2. Define the map

ϕ : Ca ⊗ Cn ⊗ Cn → Sn((Ca)∗)

T )→ det(z1X1 + · · ·+ zaXa)

where (X1, . . . , Xa) are the slices of T in the first direction.

Then ϕ is an SLa × SLn × SLn-equivariant map where the action of SLn × SLn onSn((Ca)∗) is assumed to be trivial. In particular, ϕ( maps the invariants of degree nforms in a variables to the invariants of a× n× n-tensors.

Proof. The map

ϕ : Ca ⊗ Cn ⊗ Cn → Sn((Ca)∗)

(X1, . . . , Xa) )→ det(z1X1 + · · ·+ zaXa)

satisfies

ϕ((g, h, k)·T ) = detM (g,h,k)·T (z1, . . . , za) = det(hMT (gt·z)k−1) = det(MT (gt·z)) = g·ϕ(T ).

Thus, assuming SLn×SLn acts trivially on Sn((Ca)∗), we deduce that ϕ is an equivariantmap.



Example 5.2.3. Let a = b = c = 2. The invariant ring C[Sn((C2)∗)]SL2 is generated bythe discriminant ∆, i.e. if

f = az21 + bz1z2 + cz22 ∈ S2((C2)∗)

then f is in the nullcone if and only if ∆(f) = b2 − 4ac = 0.

Let the slices of T be given by#X1 =

7T 111 T 112

T 121 T 122

8, X2 =

7T 211 T 212

T 221 T 222

8$. Then

ϕ(T ) = det(z1X1 + z2X2) = (T 111T 122 − T 112T 121)z21

+ (T 111T 222 + T 211T 122 − T 112T 221 − T 121T 212)z1z2

+ (T 211T 222 − T 212T 221)z22

so

∆(ϕ(T )) =(T 122)2(T 211)2 + (T 121)2(T 212)2 + (T 112)2(T 221)2 + (T 111)2(T 222)2

+ 4T 111T 122T 212T 221 + 4T 112T 121T 211T 222

− 2T 121T 122T 211T 212 − 2T 112T 122T 211T 221 − 2T 112T 121T 212T 221

− 2T 111T 122T 211T 222 − 2T 111T 121T 212T 222 − 2T 111T 112T 221T 222.

This is a well-known tensor invariant called Cayley hyperdeterminant and denoted byDet2. It is known that the invariant ring K[C2 ⊗ C2 ⊗ C2]SL2×SL2×SL2 is spanned byDet2. We analyze Cayley hyperdeterminant in next sections.

5.3 Invariants of Forms

In the previous section we showed that

ϕ : Ca ⊗ Cn ⊗ Cn → Sn((Ca)∗)

(Y1, . . . , Ya) )→ det(x1Y1 + · · ·+ xaYa)

is an SLa×SLn×SLn-equivariant map so the induced ring map ϕ( sends the invariantsof the space Sn((Ca)∗) to the invariants of a × n × n-tensors. Therefore, it might beuseful to study the invariants of forms and use ϕ to generate invariants of tensors.

There is a problem with the described approach. To our knowledge, a set of generatorsfor the invariant ring of Sn((Ca)∗) is known only in the following cases (for a reference,see [Stu93]):

1. a = 2 and n ≤ 8. However, for any n, a description of the nullcone of binary formsof degree n is known.

2. n = 2 and a is anything. The invariant ring of quadratic forms in any number ofvariables is spanned by the discriminant (which equals to the determinant of thecorresponding symmetric matrix).



3. a = 3 and n ≤ 4. The invariants of ternary cubics are known and called Aronholdinvariants. The invariants of ternary quartics are more involved than than thecubic case, but a complete set of generators are known (see [Dix87]).

We will study these cases in this section but we start with the most basic invariant offorms.

Definition 5.3.1. Let a and n be fixed. The discriminant∆ is an irreducible polynomialin the coefficients of a form f ∈ Sn((Ca)∗) such that ∆(f) = 0 if and only if f has anon-zero singular point, i.e. there exists p ∈ Ca − 0 which is a common zero of thepolynomials ∂f/∂zi, i = 1, . . . , a.

Remark 5.3.2. ∆ is defined only up to a scalar multiple. However, it is unique up to asign if we impose the condition that its coefficients are relatively prime integers. Observethat the existence of ∆ (over C or over Z) is not a priori clear.

Proposition 5.3.3. The discriminant has the following properties:

1. ∆ is an invariant under the action of SLa on Sn((C)a)∗).

2. deg∆ = a(n− 1)a−1.

Proof. 1. Let f ∈ Sn((Ca)∗). For g ∈ SLn, we use the chain rule to obtain

∂(g · f)∂zi

=

a+

j=1

gij∂f

∂zj.

As g is an invertible matrix, we deduce that the zero sets of partial derivatives off and partial derivatives of g · f are the same. Thus, we have

∆(f) = 0 ⇐⇒ ∆(g · f) = (g−1 ·∆)(f) = 0.

As ∆ and g−1 ·∆ have the same zero set and they have the same degree, we deducethat they must be constant multiples of each other. In other words, for all g ∈ SLa,there exists λ(g) ∈ C satisfying

g ·∆ = λ(g)∆.

Now, λ : SLa → C is a group homomorphism as

λ(gh)∆ = (gh) ·∆ = g · (h ·∆) = λ(h)(g ·∆) = λ(g)λ(h)∆

which implies that λ(gh) = λ(g)λ(h). However, as SLa has no non-trivial groupcharacters, we deduce that λ ≡ 1 so ∆ is an invariant.

2. See Chapter 9, Section 2 of [GKZ94].



The first case we consider are the binary forms. Sets of generators for the invariantring of binary forms up to degree 8 are known. Moreover, there is a characterization ofthe nullcone of binary forms independent of the degree.

Theorem 5.3.4. Let f ∈ Sn((C2)∗) be a binary form of degree n. Then the followingare equivalent

1. f ∈ NSL2(Sn((C2)∗)).

2. There exists g ∈ SL2 such that g · f is of the form

λ1 · zn1 + λ2 · zn−11 z2 + · · ·+ λn−⌊n/2⌋ · z

⌊n/2⌋+11 z

n−⌊n/2⌋−12

for some λi ∈ C.

3. There exists a non-zero singular point of f of multiplicity at least ⌊n/2⌋+ 1.

Proof. (1 ⇐⇒ 2) Consider the maximal torus

T = 7λ 00 λ−1

8| λ ∈ C×

of SL2. Then a polynomial f ∈ Sn((C2)∗) is in the nullcone of T if and only if it is ofthe form given in 2. Thus, the equivalence follows by the Hilbert-Mumford Criterion(see Theorem 2.2.34).

(2 ⇒ 3) The point (0, 1) ∈ C2 is a singular point of any polynomial in the given formwith multiplicity at least ⌊n/2⌋+ 1.

(3 ⇒ 2) Let (c1, c2) be a singular point of f of multiplicity at least ⌊n/2⌋ + 1. Letg ∈ SL2 such that

gt ·701

8=

7c1c2

8.

Consider the map

gt : P1 → P1

[x : y] )→ [g11x+ g21y : g12x+ g22y].

Then gt gives an isomorphism

gt : P1 ⊇ V (g · f) → V (f) ⊆ P1

of varieties. In particular, if (c1, c2) is a singular point of f of the given multiplicity,then (0, 1) = (gt)−1(c1, c2) is a singular point of g · f of the given multiplicity. Now, tofinish the proof, we observe that (0, 1) is a singular point of a binary form f of the givenmultiplicity if and only if it is of the given form in 2.



To our knowledge, in the case of ternary forms (i.e. a = 3), the only case where acomplete set of generators for the invariant ring is the case of ternary cubics. We givethe following characterization of the invariant ring. For the proof, see [Stu93].

Theorem 5.3.5. The invariant ring C[S3((C3)∗)]SL3 is spanned by two invariants Sand T (called the Aronhold invariants) of degree 4 and 6 respectively. Moreover, wehave ∆ = 4S3 − T 2.

Remark 5.3.6. Recall that for a ternary cubic f ∈ S3((C3)∗), ∆(f) = 0 if and only if fhas a singular point. If furthermore both S and T vanishes at f , then this singularityis a cuspidal singularity (see the following proposition). For a ternary cubic f , f has acuspidal singularity if and only if there exists g ∈ SL3 such that

g · f = F (z1, z2) + z3G(z1, z2)

for a binary cubic F and a binary quadric G satisfying G = (az1 + bz2)2.

Proposition 5.3.7. Let f ∈ Sn((C3)∗) be a ternary form of degree n ≥ 3. If f is in thenullcone, then there exists g ∈ SL3 such that g · f is of the form

g · f = Pn(z1, z2) + Pn−1(z1, z2)z3 + · · ·+ P⌊n/3⌋+1(z1, z2)zn−⌊n/3⌋−13

where each Pi is a binary form of degree i and P⌊n/3⌋+1 = (az1 + bz2)⌊n/3⌋+1 for some

a, b ∈ C.

Proof. By the Hilbert-Mumford criterion, there exists g ∈ SL3 such that g · f is in thenullcone of the torus. Let

h(t) =

%

'ta1

ta2

ta3

(

* ∈ SL3

be a one-parameter subgroup annihilating g · f . If (a1, a2, a3) = (0, 0, 0), then the one-parameter subgroup is trivial which implies that f = 0. In this case, we are done asP⌊n/3⌋+1 = 0 satisfies the condition. Hence, we may assume that (a1, a2, a3) ∕= (0, 0, 0).Without loss of generality, assume that a3 = mina1, a2, a3 so in particular, a3 < 0.Write

g · f = Pn(z1, z2) + · · ·+ P2(z1, z2)zn−23 + P1(z1, z2)z

n−13 + λzn3 .

Then, each monomial zp1zq2z

r3 of h(t) ·(g ·f) is multiplied with the coefficient tpa1+qa2+ra3 .

Observe that for d ≤ ⌊n/3⌋ and i ≤ d we have

ia1+(d− i)a2+(n− d)a3 ≤ ia1+(d− i)a2+(d− i)a1+ ia2+ da3 ≤ d(a1+ a2+ a3) = 0.

As h(t) annihilates g · f , the above equation implies that the coefficient of zi1zd−i2 zn−d

3

must be 0 in g · f . Thus, for d ≤ ⌊n/3⌋, we have Pd = 0. Now, set d = ⌊n/3⌋ + 1. Fori = 1, . . . , d− 1, we have

ia1 + (d− i)a2 + (n− d)a3 ≤ maxi, d− ia1 +maxi, d− ia2 +maxi, d− ia3 = 0



so the coefficients of zi1zd−i2 in Pd are 0. Moreover,

(da1 + (n− d)a3) + (da2 + (n− d)a3) ≤ da1 + da2 + da3 = 0.

Thus, at least one of the monomials zd1zn−d3 or zd2z

n−d3 must have coefficient equal to zero

which implies that Pd is a scalar multiple of either zd1 or zd2 . We obtained the desiredresult.

5.4 Tensor Invariants

Definition 5.4.1. Let a, b, c ∈ Z≥1 and pick bases x1, . . . , xa of Ca, y1, . . . , yb of Cb andz1, . . . , zc of Cc. Denote the corresponding dual bases with xi, yj , zk. We call a tensor

T : (Ca)∗ × (Cb)∗ × (Cc)∗ → C

degenerate if the system∂T

∂xi=

∂T

∂yj=

∂T

∂zk= 0

has a solution (α,β, γ) ∈ (Ca)∗ × (Cb)∗ × (Cc)∗ with α ∕= 0,β ∕= 0, γ ∕= 0. In this case,(α,β, γ) is a singular point of the map T and we call such a singular point (α,β, γ) anon-trivial singular point.

Remark 5.4.2. Observe that as T is multilinear, any point (α,β, γ) ∈ (Ca)∗×(Cb)∗×(Cc)∗

with one of α,β, γ equal to zero is a singular point. Thus we use the term non-trivialsingular point for a singular point if all α,β, γ is non-zero.

In the special case that the maximum of a−1, b−1, c−1 is less than or equal to the sumof the other two, the set of degenerate tensors is a hypersurface, given as the zero set ofCayley’s hyperdeterminant. The following definition can be taken as a theorem:

Definition 5.4.3. Let a, b, c ∈ Z≥1 be such that the maximum of a − 1, b − 1, c − 1 isless than or equal to the sum of other two. The Cayley hyperdeterminant Det of format(a− 1, b− 1, c− 1) is an irreducible polynomial

Det : Ca ⊗ Cb ⊗ Cc → C

such that Det(T ) = 0 if and only if T is degenerate.

Remark 5.4.4. Observe that as in the case of the discriminant ∆, Det is defined onlyup to a scalar multiple. However, we can impose the condition that its coefficients arerelatively prime integers and then Det is unique up to a sign.

In [GKZ94], Gelfand, Kapranov and Zelevinsky give formulas for the degree of Det.For the special case in which we are interested, we have the following.

Proposition 5.4.5 ([GKZ94], Chapter 14, Section 2B). 1. The degree of the hyper-determinant of format (1, n− 1, n− 1) is 2n(n− 1).



2. The degree of the hyperdeterminant of format (2, n− 1, n− 1) is 3n(n− 1)2.

3. The degree of the hyperdeterminant of format (3, n − 1, n − 1) is 23n(n − 1)(n −

2)(5n− 3).

We give a classification of degenerate tensors in terms of coordinates:

Theorem 5.4.6. Let T ∈ Ca ⊗ Cb ⊗ Cc. Then the following are equivalent.

1. T is degenerate, i.e. T : (Ca)∗ × (Cb)∗ × (Cc)∗ → C has a non-trivial singularpoint.

2. There exists (α,β, γ) ∈ (Ca)∗ × (Cb)∗ × (Cc)∗ such that all α,β, γ ∕= 0 and

∀ψ ∈ (Ca)∗, T (ψ,β, γ) = 0, ∀ψ ∈ (Cb)∗, T (α,ψ, γ) = 0, ∀ψ ∈ (Cc)∗, T (α,β,ψ) = 0.

3. There exists (g, h, k) ∈ SLa × SLb × SLc such that the slices of (g, h, k) · T are ofthe form

Y1 =

%

&&&&'

∗ ∗ . . . ∗

∗ . . ....

.... . . ∗

∗ . . . ∗ 0

(

))))*, . . . , Ya−1 =

%

&&&&'

∗ ∗ . . . ∗

∗ . . ....

.... . . ∗

∗ . . . ∗ 0

(

))))*, Ya =

%

&&&&'

∗ . . . ∗ 0...

. . . 0

∗ ∗...

0 0 . . . 0

(

))))*

Proof. (1 ⇒ 2) Let (α,β, γ) be a non-trivial singular point of T . We show that ∀ψ ∈(Ca)∗, T (ψ,β, γ) = 0. The proofs of other two equalities are similar.

Let ψ ∈ (Ca)∗. As T is multilinear, we have

T (α+ ψ,β, γ) = T (α,β, γ) + T (ψ,β, γ) = T (ψ,β, γ).

On the other hand, we have the Taylor expression of T :

T (α+ ψ,β, γ) = T (α,β, γ) +

a+

i=1

∂T

∂xi|(α,β,γ) (ψ,β, γ) + higher order derivatives.

Note that the first summand is zero as (α,β, γ) is a singular point, the secomd summandis zero as the first partial derivatives vanish and the third summand is zero as T ismultilinear : the degree of each variable xi is 1 so the second partial derivatives vanish.Thus we obtain T (ψ,β, γ) = 0.

(2 ⇒ 1) Let (α,β, γ) satisfy the statement. Then for ψ ∈ (Ca)∗ we have

DψT = limε→0

T (α+ εψ,β, γ)− T (α,β, γ)

ε= 0

where the last equality follows by the fact that T (·,β, γ) ≡ 0. Thus the partial derivatives∂T∂xi vanish at (α,β, γ). The partial derivatives for yj and zk similary follow.



(2 ⇒ 3) Let (α,β, γ) satisfy the statement. As each α,β, γ is non-zero, there existg ∈ SLa, h ∈ SLb, k ∈ SLc such that

g · α = xa, h · β = yb, k · γ = zc.

Here, xi, yj , zk denote the standard dual basis vectors of Ca,Cb,Cc, respectively. Now,setting F = (g, h, k) · T , we have

∀ψ ∈ (Ca)∗, F (ψ, yb, zc) = T (g−1ψ,β, γ) = 0.

Similarly we have F (xa,ψ, zk) = 0 and F (xa, yb,ψ) = 0. Observe that if (Y1, Y2, . . . , Ya)are the slices of F in the first direction, then

∀i = 1, . . . , a, 0 = F (xi, yb, zc) = (zc)t · Yi · yb = (Yi)cb

∀j = 1, . . . , b, 0 = F (xa, yj , zc) = (zc)t · Ya · yj = (Ya)cj

∀k = 1, . . . , c, 0 = F (xa, yb, zk) = (zk)t · Ya · yb = (Ya)kb.

These equations show that (Y1, . . . , Ya) is of the expected form.

(3 ⇒ 2) Let the slices of F = (g, h, k) · T in the first direction be of the given form.Then F satisfies

∀ψ ∈ (Ca)∗, F (ψ, yb, zc) = 0, ∀ψ ∈ (Cb)∗, F (xa,ψ, zc) = 0, ∀ψ ∈ (Cc)∗, F (xa, yb,ψ) = 0.

Thus setting α = g−1xa,β = g−1yb, γ = g−1zk, T satisfies the given property.

Corollary 5.4.7. Degeneracy of a tensor T is preserved under the action of SLa ×SLb × SLc. In particular, if Det exists then it is an invariant.

Proof. The first part of the corollary follows by the equivalence of 1 and 3 in the previoustheorem. For the second part, we first observe that for all (g, h, k) ∈ SLa × SLb × SLc

Det = 0 = (g, h, k) ·Det = 0.

Thus Det and (g, h, k) · Det are constant multiples of each other. As in the proof of ∆being an invariant of forms, the result follows by the fact that SLa × SLb × SLc has nonon-trivial group characters.

Observe that both Det and ∆ gives information about the singular points of elementsin corresponding spaces. Using the map

φ : Ca ⊗ Cn ⊗ Cn → Sn((Ca)∗)

(X1, . . . , Xa) )→ det(z1X1 + · · ·+ zaXa)

from tensors to forms, we give the following relation between Det and ∆.

Theorem 5.4.8. If T ∈ Ca ⊗ Cn ⊗ Cn is degenerate, then ∆(ϕ(T )) = 0.



Proof. Using Theorem 5.4.6 above, we may assume that the slices (X1, . . . , Xa) of T sat-isfy

X1 =

%

&&&&'

∗ ∗ . . . ∗

∗ . . ....

.... . . ∗

∗ . . . ∗ 0

(

))))*, . . . , Xa−1 =

%

&&&&'

∗ ∗ . . . ∗

∗ . . ....

.... . . ∗

∗ . . . ∗ 0

(

))))*, Xa =

%

&&&&'

∗ . . . ∗ 0...

. . . 0

∗ ∗...

0 0 . . . 0

(

))))*

A direct computation shows that

ϕ(T ) = det(z1X1+· · ·+zaXa) = det(Xa)·zna+a−1+

i=1

λi·zn−1a zi+terms with smaller za degree

where

λi =

c+

j=1

b+

k=1

(Xi)jk ·Mjk

and Mjk equals to the maximal minor of Xa obtained by deleting j-th row and k-thcolumn. Now, detXa = 0 so the coefficient of zna is zero. Moreover, only non-zero minorof Xa is Mcb but in this case (Xi)cb = 0 so for all i we also have λi = 0. Thus (0, . . . , 0, 1)is a singular point of ϕ(T ) so ∆(ϕ(T )) = 0.

Corollary 5.4.9. Let a = 2 or 3. Then ϕ(∆ = Det.

Proof. By the theorem we have Det = 0 ⊆ ϕ(∆ = 0 so Det divides ϕ(∆. Recallfrom Proposition 5.3.3 that for a = 2 we have deg∆ = 2(n − 1). As the coefficientsof the form ϕ(T ) are of degree n in the entries of T , we obtain degϕ(∆ = 2n(n − 1).Moreover, using Proposition 5.4.5, we get

degϕ(∆ = 2n(n− 1) = degDet

so ϕ(∆ = Det.

When a = 3 we have deg∆ = 3(n− 1)2 so

degϕ(∆ = 3n(n− 1)2 = degDet

and once again we obtain ϕ(∆ = Det.

Remark 5.4.10. When a = 4, the result no longer holds as

degϕ(∆ = 4n(n− 1)3 > degDet .

A natural question is whether Im(ϕ() = C[Sn((Ca)∗)]SLm×SLn×SLn holds. This is notthe case as the following example shows.



Example 5.4.11. Let T ∈ C3 ⊗ C3 ⊗ C3 be the tensor with slices

X1 =

%

'0 0 00 0 10 −1 0

(

* , X2 =

%

'0 0 −10 0 01 0 0

(

* , X3 =

%

'0 1 0−1 0 00 0 0

(

*

Then we haveϕ(T ) = det(z1X1 + z2X2 + z3X3) = 0

so in particular every invariant of forms vanishes at ϕ(T ). However, T is not in thenullcone NSL3×SL3×SL3(C3 ⊗ C3 ⊗ C3). To see why that is true, we first make thefollowing observation:

Claim : For any g ∈ SL3 we have

(g, g, g) · T = T.

Proof. Let g ∈ SL3. Consider the matrix

A =

%

'x1 x2 x3

y1 y2 y3

z1 z2 z3

(

*

and consider det(A) as a multilinear function

det(A) : C3 × C3 × C3 → C

where we set xi as the standard basis vectors of the first factor C3 and thus xi act asthe standard dual basis vectors. Similarly, we set yi and zi as the standard basis vectorsof the second and the third factors respectively.

Then for i, j, k ∈ [3] we have

T (xi, yj , zk) = ytj ·Xi · zk = (Xi)jk

and a direct computation shows that

det(A)(xi, yj , zk) =

;0 if i = j or i = k or j = k

sgn(1 )→ i, 2 )→ j, 3 )→ k) otherwise.

Observe that this function exactly equals (Xi)jk. Therefore we have

T = det(A).

Then for g ∈ SL3 we obtain

(g, g, g) · T = (g, g, g) · det(A) = det(Ag−1) = det(A) det(g−1) = det(A) = T

so the claim holds.



This observation is enough to show that T is not in the nullcone: To reach a contradic-tion, assume that we have a one-parameter subgroup (g(t), h(t), k(t)) ⊆ SL3×SL3×SL3

that annihilates T . Then we have

0 = limt→0

(g(t), h(t), k(t)) · T

= limt→0

(I3, h(t)g(t)−1, k(t)g(t)−1)

#(g(t), g(t), g(t)) · T

$

= limt→0

(I3, h(t)g(t)−1, k(t)g(t)−1)T

which implies that

limt→0

#h(t)g(t)−1

$#(Y1, Y2, Y3)

$#g(t)k(t)−1

$= 0.

On the other hand, we know that (Y1, Y2, Y3) is not in the nullcone of the left-right actionby Example 3.2.15 of Chapter 3.

The above example is not very surprising as in Chapter 3 we see that the nullcone ofthe left-right action is not spanned by det(t1X1 + · · ·+ tmXm).

5.5 The Computation of the Nullcone of Some Spaces

Recall that the map

ϕ : Ca ⊗ Cn ⊗ Cn → Sn((Ca)∗)

(X1, . . . , Xa) )→ det(z1X1 + · · ·+ zaXa)

is SLa × SLn × SLn-equivariant. Thus, ϕ maps the nullcone of the order three tensorsto the nullcone of forms.

5.5.1 Two Slices Tensors : C2 ⊗ Cn ⊗ Cn

We start with a classification of the nullcone of C2 ⊗ Cn ⊗ Cn in terms of ϕ:

Theorem 5.5.1. Let T ∈ C2 ⊗ Cn ⊗ Cn. Then T is in the nullcone if and only if thebinary form ϕ(T ) is in the nullcone.

Proof. The only if part is evident. We prove the if part. We assume that T ∈ C2⊗Cn⊗Cn

and the binary form ϕ(T ) is in the nullcone.

Recall the characterization of the nullcone of binary forms from Theorem 5.3.4: Abinary form f(z1, z2) is in the nullcone if and only if there exists g ∈ SL2 such that g · fis of the form

g · f = λ1 · zn1 + · · ·+ λn−⌊n/2⌋ · z⌊n/2⌋+11 z

n−⌊n/2⌋−12 (5.2)



for some λi ∈ C.

Thus, by multiplying g with T , we assume that ϕ(T ) is of the above form. Let(X,Y ) be the slices of T in the first direction. As the coefficient of zn2 in ϕ(T ) =det(z1X + z2Y ) equals det(Y ), we have det(Y ) = 0. Thus, there exist g, h ∈ SLn suchthat

gY h−1 =

7Ir 0

0 0

8

where r = rk(Y ) < n. Let

gXh−1 =

7A B

C D

8

Here, A is an r × r-matrix, B is an r × (n − r)-matrix, C is an (n − r) × r-matrix andD is an (n− r)× (n− r)-matrix. We update T ← (I2, g, h) · T .

If 2r ≤ n, then Y has too many zeroes and T must be in the nullcone: We set

G(t) =

7t2 00 t−2

8, H(t) =

%

&&&&&&&&&&&&&&&'

t. . .

tt−1

. . .

t−1

1. . .

1

(

)))))))))))))))*

= K(t)−1

where H and K has r many t’s, r many t−1’s and n − 2r (possibly 0) many 1’s. Thena direct computation shows that no entry of (G,H,K) · T contains a negative exponentof t : A is multiplied with t4, the entries of B and C are multiplied with either t2 or t3,the entries of D are multiplied with either t2, t or 1 and lastly the entries of Y are allmultiplied with 1. Thus the limit

F = limt→0

(G(t), H(t),K(t)) · T

exists and satisfies

ϕ(F ) = ϕ(limt→0

(G(t), H(t),K(t)) · T ) = limt→0

G(t) · ϕ(T ) = 0.

By Lemma 3.2.5, ϕ(F ) = 0 implies that the slices of F is in the nullcone of the left-rightaction: There exist g(t), h(t) ∈ SLn such that

limt→0

(I2, g(t), h(t)) · F = 0.

As T contains a tensor F in its orbit closure that is in the nullcone, T itself must be inthe nullcone. Thus, the theorem holds if 2r ≤ n.



Now we assume that 2r > n. Observe that as ϕ(T ) is of the given form in (5.2) above,the coefficient of zr2 must be zero. On the other hand, a direct computation of ϕ(T )shows that this coefficient equals det(D) so we obtain det(D) = 0. Let p, q ∈ SLn−r

such that pDq−1 is the block matrix

pDq−1 =

70 0

0 Ik

8

where k = rk(D) < n− r. Set

g =

7Ir 0

0 p

8, h =

7Ir 0

0 q

8.

Then we have

gXh−1 =

7Ir 0

0 p

8·7A BC D

8·7Ir 0

0 q

8−1

=

%

'A B1 B2

C1 0 0

C2 0 Ik

(

* , gY h−1 =

%

'Ir 0 0

0 0 0

0 0 0

(

*

We update T ← (I2, g, h) · T so we can assume that the slices (X,Y ) of T are of thegiven form above. Moreover, by some row and column operations, we may assume thatB2 and C2 are 0: We can add some multiples of the rows of Ik to B2 and some multiplesof the columns of Ik to C2 and make every entry equal to 0. This does not change Y asthe entries of Y that corresponds to the entries of Ik are all 0.

Let b = n − r − k be the size of the 0 block matrix in the middle of the matrices Xand Y .

We may assume that that the rows of C1 are linearly independent: If not, then wecan apply some row operations to X and set the last row of C1 to be 0. But the rows ofY corresponding to the rows of C1 are all 0, so if we apply these row operations to bothX and Y , then they share a common row. In this case T is in the nullcone. Similarly, wemay assume that the columns of B1 are linearly independent. Hence, there exist somematrices p, q ∈ SLb such that

pC1 =1λIb 0

2, B1q

−1 =

7µIb0

8

for some non-zero λ, µ ∈ C. Set

g =

%

'Ir 0 0

0 p 0

0 0 Ik

(

* , h =

%

'Ir 0 0

0 q 0

0 0 Ik

(

*

Then we have

gXh−1 =

%

&&'

A1 A2 µIb 0

A3 A4 0 0

λIb 0 0 0

0 0 0 Ik

(

))* , gY h−1 =

%

&&'

Ib 0 0 0

0 Ir−b 0 0

0 0 0 0

0 0 0 0

(

))*



Update T ← (I2, g, h) · T . We can assume that A1, A2, A3 are all 0 as we can add rowsof λIb and columns of µIb to A1, A2, A3 to make all entries 0. This operation does notchange Y as the entries of Y that corresponds to the entries of λIb and µIb are 0. Thus,we can write

z1X + z2Y =

%

&&'

z2Ib 0 z1µIb 0

0 z1A4 + z2Ir−b 0 0

z1λIb 0 0 0

0 0 0 z1Ik

(

))*

so

ϕ(T ) = det(z1X + z2Y ) = zk+2b1 (zr−b

1 det(A4) + · · ·+ z1zr−b−12 Tr(A4) + zr−b

2 ).

On the other hand, in the process of reducing (X,Y ) to this form, we did not use theaction of SL2, i.e. whenever we updated T with some element (G,H,K) ∈ SL2×SLn×SLn, we had G = I2. Thus, ϕ(T ) still is of the form given in (5.2) which implies that2(r − b) < n (since the coefficient of zr−b

2 is non-zero). Now, the equations

2(r − b) < n

r + b+ k = n

together imply that

k > r − 3b.

Set

G(t) =

%

&&'

tIr 0 0 0

0 t−3Ib 0 0

0 0 t−1Ir−3b 0

0 0 0 Ik−(r−3b)

(

))* = H(t)−1

and observe that det(G(t)) = det(H(t)) = tr−3b−(r−3b) = 1 so G(t), H(t) ∈ SLn. Thena direct computation shows that the limit

F = limt→0

(

7t2 00 t−2

8, G(t), H(t)) · T

exists: the entries of A4 are multiplied with t2 · t · t = t4, entries of λIb and µIb aremultiplied with t2 · t · t−3 = 1, entries of Ik are multiplied with either t2 or 1 and lastlythe entries of Y are multiplied with t−2 · t · t = 1. Moreover, we have

ϕ(F ) = limt→0

7t2 00 t−2

8· ϕ(T ) = 0.

Thus, F is in the nullcone of the left-right action (by Lemma 3.2.5) and thus T alsobelongs to the nullcone.



5.5.2 Cuboids : C3 ⊗ C3 ⊗ C3

In the previous section, we have proved that a tensor T ∈ C2⊗Cn⊗Cn is in the nullconeif and only if the form ϕ(T ) is in the nullcone of forms. This no longer holds for tensorsT ∈ U ⊗ V ⊗ W with dimU, dimV, dimW ≥ 3. One example is given in Example5.4.11.

Compare this phenomena with Lemma 3.2.5: For pairs (X1, X2) ∈ Mat2n,n matrices,(X1, X2) is in the nullcone of the left-right action if and only if det(sX1 + tX2) ≡ 0.This is not true for triples of matrices. Thus, we need some further assumptions on T .In the case that n = 3, we can produce such a condition. In the literature, the elementsof C3 ⊗ C3 ⊗ C3 are called cuboids.

Definition 5.5.2. Let T ∈ C3 ⊗ C3 ⊗ C3 be a cuboid. Let (A,B,C) be the slices of Tin the first direction. We define

ε1 = 2[x : y : z] ∈ P2 | rk(xA+ yB + zC) ≤ 1

where 2 denotes the cardinality of the set. Note that we possibly have ε1 = ∞. Similarly,if (A′, B′, C ′), (A′′, B′′, C ′′) are the slices of T in the second and the third directionrespectively, then we define

ε2 = 2[x : y : z] ∈ P2 | rk(xA′ + yB′ + zC ′) ≤ 1,ε3 = 2[x : y : z] ∈ P2 | rk(xA′′ + yB′′ + zC ′′) ≤ 1.

Remark 5.5.3. Let T = [T ijk]i∈[3],j∈[3],k∈[3] be a cuboid and let (A,B,C) ∈ Mat33,3 be

its slices in the first direction. Let S = [T kji]i∈[3],j∈[3],k∈[3] and R = [T jik]i∈[3],j∈[3],k∈[3].Then the slices of T in the second direction are the slices of S in the first direction andthe slices of T in the third direction are the slices of R in the first direction.

Theorem 5.5.4. Let T ∈ C3 ⊗ C3 ⊗ C3 and let S and R be cuboids given in the aboveremark. Then T is in the nullcone if and only if

1. ϕ(T ) = det(z1A + z2B + z3C),ϕ(S),ϕ(R) ∈ S3((C3)∗) are all in the nullcone ofternary cubics and

2. (ε1, ε2, ε3) ∕= (0, 0, 0).

Proof. (⇒) Assume that T is in the nullcone. Since ϕ is an equivariant map, ϕ(T ) is inthe nullcone of ternary cubics. S,R are also in the nullcone as if

limt→0

(g(t), h(t), k(t)) · T = 0,

then we have

limt→0

(k(t), h(t), g(t)) · S = 0,

limt→0

(h(t), g(t), k(t)) ·R = 0.



Thus, ϕ(S) and ϕ(R) are also in the nullcone of ternary cubics.

By Remark 5.3.6, there exists g ∈ SL3 such that

ϕ(T ) = det(z1A+ z2B + z3C) = f(z1, z2) + λz21z3

for some binary cubic f and λ ∈ C. In particular, we have det(C) = 0. If rk(C) ≤ 1,then we are done. Hence, we assume that rk(C) = 2. Let g, h ∈ SL3 such that

gCh−1 =

%

'1 0 00 1 00 0 0

(

* .

Replace T with (I3, g, h) · T (since the first matrix is I3, this does not change ϕ(T )).Note that the coefficient of z23 in ϕ(T ) is z1A33 + z2B33. Thus, we have A33 = B33 = 0.Thus, we have

T =

3%

'A11 A12 A13

A21 A22 A23

A31 A32 0

(

* ,

%

'B11 B12 B13

B21 B22 B23

B31 B32 0

(

* ,

%

'1 0 00 1 00 0 0

(

*4

S =

3%

'A11 B11 1A21 B21 0A31 B31 0

(

* ,

%

'A12 B12 0A22 B22 1A32 B32 0

(

* ,

%

'A13 B13 0A23 B23 00 0 0

(

*4

R =

3%

'A11 A12 A13

B11 B12 B13

1 0 0

(

* ,

%

'A21 A22 A23

B21 B22 B23

0 1 0

(

* ,

%

'A31 A32 0B31 B32 00 0 0

(

*4.

The coefficient of z22z3 in ϕ(T ) is −B23B32 − B13B31 and the coefficient of z1z2z3 inϕ(T ) is −(A23B32 +A32B23 +A13B31 +A31B13). We obtain

B23B32 +B13B31 = 0

A23B32 +A32B23 +A13B31 +A31B13 = 0.(5.3)

Now, if both B13 and B23 are zero, then the third slice of S has rank ≤ 1 and we obtainthe desired result (ε2 ∕= 0). Thus, without loss of generality, assume that B13 ∕= 0 (thecase B31 ∕= 0 will be similar). By adding −B23/B13 times the first row to the secondrow, we can make B23 = 0. Then, by (5.3), we have B31 = 0. If B32 is also zero, then thethird slice of R has rank ≤ 1. Thus, we may assume that B32 ∕= 0. Since B31 = B23 = 0,by (5.3) we have A23B32 + A31B13 = 0. If A23 = 0, then since B23 = 0, the third sliceof S has rank ≤ 1. Thus, we assume that A23 ∕= 0. We add −A13/A23 times the secondrow to the first row so A13 = 0. Similarly, if A31 = 0, then the third slice of R has rank≤ 1. Thus, we assume that A31 ∕= 0. By adding −A32/A31 times the first column to thesecond column, we assume that A32 = 0. In the end of these operations, T is of the form

T =

3%

'∗ ∗ 0∗ ∗ A23

A31 0 0

(

* ,

%

'∗ ∗ B13

∗ ∗ 00 B32 0

(

* ,

%

'1 −A13/A23 −A32/A31 0

−B23/B13 1 + (B23/B13) · (A32/A31) 00 0 0

(

*4



Note that some new entries appeared in the third slice as a result of the row and columnoperations we applied in the process. Consider the one-parameter subgroup

t )→ G(t) =

3%

't

tt−2

(

* ,

%

't

tt−2

(

* ,

%

't−1

t−1

t2

(

*4

Set F = limt→0G(t) · T . Then we have

F =

3%

'0 0 00 0 A23

A31 0 0

(

* ,

%

'0 0 B13

0 0 00 B32 0

(

* ,

%

'1 −A13/A23 −A32/A31 0

−B23/B13 1 + (B23/B13) · (A32/A31) 00 0 0

(

*4

Now, ϕ(F ) = limt→0

%

't

tt−2

(

*·ϕ(T ) = λz21z3. On the other hand, direct computation

shows that

0 = ϕ(F ) = (−A13A31−A23A32)z21z3+(−A31B13−A32B23−A23B32)z1z2z3−(B23B32)z

22z3.

Thus, B23B32 = 0 which implies that B23 = 0. Moreover, (5.3) implies that the co-efficient of z1z2z3 = 0 so ϕ(F ) = 0. Lastly, −A13A31 − A23A32 = 0 implies that−A13/A23 −A32/A31 = 0. Thus,

F =

3%

'0 0 00 0 A23

A31 0 0

(

* ,

%

'0 0 B13

0 0 00 B32 0

(

* ,

%

'1 0 00 1 00 0 0

(

*4.

Swap the second columns with the first columns and multiply the first columns with −1to obtain 3%

'0 0 00 0 A23

0 A31 0

(

* ,

%

'0 0 B13

0 0 0−B32 0 0

(

* ,

%

'0 1 0−1 0 00 0 0

(

*4.

Using the equation A31B13+A23B32 = 0, we deduce that this tensor is in the same orbitwith the tensor given in Example 5.4.11 (which is not in the nullcone), we deduce thatF is not in the nullcone. Since T contains F in its orbit closure, T is not in the nullcone,contradicting the assumption. Thus, (ε1, ε2, ε3) ∕= (0, 0, 0).

(⇐) Since the condition is symmetric, we may assume that ε1 ∕= 0.

By acting in the first direction, we can assume that rk(C) ≤ 1. If C = 0, then let

G(t) =

3%

't

tt−2

(

* , I3, I3

4.



Thenlimt→0

G(t) · T = 0

so T is in the nullcone. Thus, we assume that rk(C) = 1. There exists g, h ∈ SL3 suchthat

gCh−1 =

%

'1 0 00 0 00 0 0

(

* .

By replacing T with (I3, g, h) · T , we may assume that C is equal to the above matrix.Let

A =

7A22 A23

A32 A33

8, B =

7B22 B23

B32 B33

8.

Thenϕ(T ) = det(z1A+ z2B + z3C) = p(z1, z2) + z3q(z1, z2)

wherep(x, y) = det(xA+ yB), q(x, y) = det(xA+ yB).

Recall Remark 5.3.6: Since ϕ(T ) is in the nullcone, the singular point (0, 0, 1) of ϕ(T )must be a cusp, i.e. q(z1, z2) must be a perfect square. By acting in the first directionif necessary, we may assume that q(z1, z2) = λz21 for some λ ∈ C. Hence, det(B) = 0. IfB = 0, then consider the one-parameter subgroup

G(t) =

3%

't2

t−1

t−1

(

* ,

%

't2

t−1

t−1

(

* ,

%

't2

t−1

t−1

(

*4.

Then, the limit limt→0G(t) · T = F exists: A direct computation shows that every non-zero entry of G(t) · T is of the form Tijkt

l for some l ∈ Z>0. Moreover, observe thatC11 is multiplied with t3 so if (A′, B′, C ′) are the slices of F in the first direction, thenC ′ = 0. Thus, F is in the nullcone and as T contains F in its orbit closure, T is also inthe nullcone.

Hence, we assume that B ∕= 0. There exist g, h ∈ SL2 such that

gBh−1 =

71 00 0

8.

Set

g =

%

'1 0 00 g11 g120 g21 g22

(

* , h =

%

'1 0 0

0 h11 h120 h21 h22

(

* .

and replace T with (I3, g, h) · T we can assume that the slices (A,B,C) are of the form

(A,B,C) =

3%

'∗ ∗ ∗∗ ∗ ∗∗ ∗ 0

(

* ,

%

'∗ ∗ ∗∗ 1 0∗ 0 0

(

* ,

%

'1 0 00 0 00 0 0

(

*4.



Here, A33 = 0 since it is equal to the coefficient of z1z2 in q(z1, z2) = det(z1A + z2B)which was assumed to be 0. Let

G(t) =

3%

't2

tt−3

(

* ,

%

't2

1t−2

(

* ,

%

't2

1t−2

(

*4.

Once again, F = limt→0G(t) · T exists and the slices of F are of the form

(A′, B′, C ′) =

3%

'0 0 00 0 ∗0 ∗ 0

(

* ,

%

'0 0 00 0 00 0 0

(

* ,

%

'0 0 00 0 00 0 0

(

*4.

Hence, F is in the nullcone, which implies that T is in the nullcone.


Bibliography

[ABW82] Kaan Akın, David Buchsbaum, and Jerzy Weyman. Schur Functors and SchurComplexes. Advances in Mathematics, 44(3):207 – 278, 1982.

[Art69] Michael Artin. On Azumaya Algebras and Finite Dimensional Representationsof Rings. Journal of Algebra, 11(4):532 – 563, 1969.

[BD06] Matthias Burgin and Jan Draisma. The Hilbert Null-cone on Tuples of Matricesand Bilinear Forms. Mathematische Zeitschrift, 254(4):785–809, Dec 2006.

[Coh71] P. M. Cohn. The Embedding of Firs in Skew Fields. Proceedings of the LondonMathematical Society, s3-23(2):193–213, 10 1971.

[Dix87] Jacques Dixmier. On the Projective Invariants of Quartic Plane Curves. Ad-vances in Mathematics, 64(3):279 – 304, 1987.

[DK02] Harm Derksen and Gregor Kemper. Computational Invariant Theory. Ency-clopaedia of Mathematical Sciences. Springer Berlin Heidelberg, 2002.

[DM15] Harm Derksen and Visu Makam. Polynomial Degree Bounds for Matrix Semi-Invariants. Dec 2015. arXiv:1512.03393v1 [math.RT].

[DM16a] Harm Derksen and Visu Makam. Degree Bounds for Semi-invariant Rings ofQuivers. 2016.

[DM16b] Harm Derksen and Visu Makam. Generating Invariant Ring of Quivers inArbitrary Characteristic. Oct 2016. arXiv:1610.06617 [math.RT].

[DM18a] Harm Derksen and Visu Makam. Algorithms for Orbit Closure Separa-tion for Invariants and Semi-invariants of Matrices. arXiv e-prints, pagearXiv:1801.02043, Jan 2018.

[DM18b] Harm Derksen and Visu Makam. Weyl’s Polarization Theorem in PositiveCharacteristic. Mar 2018. arXiv:1803.03602v2 [math.RT].

[Dom00] Matya Domokos. Relative Invariants of 3 × 3 Matrix Triples. Linear andMultilinear Algebra, 47(2):175–190, 2000.

[Don85] Stephen Donkin. Rational Representations of Algebraic Groups. Springer-Verlag Berlin Heidelberg, 1985. ISBN 978-3-540-15668-0.

[Don87] Stephen Donkin. On Schur Algebras and Related Algebras, II. Journal ofAlgebra, 111(2):354 – 364, 1987.


[Don88] Stephen Donkin. Invariants of unipotent radicals. Mathematische Zeitschrift,198(1):117–125, Mar 1988.

[Don92] Stephen Donkin. Invariants of Several Matrices. Inventiones mathematicae,110, 1992.

[Don93] Stephen Donkin. On tilting modules for algebraic groups. MathematischeZeitschrift, 212(1):39–60, Jan 1993.

[Don94] Stephen Donkin. Polynomial invariants of representations of quivers. Com-mentarii Mathematici Helvetici, 69(1):137–141, Dec 1994.

[DW00] Harm Derksen and Jerzy Weyman. Semi-Invariants of Quivers and Saturationfor Littlewood-Richardson Coefficients. Journal of the American MathematicalSociety, 13(3):467–479, 2000.

[DW17] Harm Derksen and Jerzy Weyman. An Introduction to Quiver Representations.American Mathematical Society, 2017. ISBN 9781470425562.

[Eis95] David Eisenbud. Commutative Algebra: With a View Toward Algebraic Ge-ometry. Graduate Texts in Mathematics. Springer, 1995.

[FH91] William Fulton and Joe Harris. Representation Theory : A First Course.Springer, 1991.

[FR] Marc Fortin and Christopher Reutenauer. Commutative/noncommutativeRank of Linear Matrices and Subspaces of Matrices of Low Rank. SeminaireLotharingien de Combinatoire.

[GKZ94] Israel M. Gelfand, Mikhail M. Kapranov, and Andrei V. Zelevinsky. Discrim-inants, Resultants, and Multidimensional Determinants. Birkhauser, Boston,MA, 1994.

[Hum75] James E. Humphreys. Linear Algebraic Groups. Springer-Verlag New York,1975. ISBN 9780387901084.

[IQKS13] Gabor Ivanyos, Youming Qiao, Marek Karpinski, and Miklos Santha. Gener-alized Wong Sequences and Their Applications to Edmonds’ Problems. CoRR,abs/1307.6429, 2013.

[IQS15a] Gabor Ivanyos, Youming Qiao, and K.V. Subrahmanyam. Constructive Non-commutative Rank Computation in Deterministic Polynomial Time OverFields of Arbitrary Characteristics. CoRR, abs/1512.03531, 2015.

[IQS15b] Gabor Ivanyos, Youming Qiao, and K.V. Subrahmanyam. Non-commutativeEdmonds’ Problem and Matrix Semi-invariants. CoRR, abs/1508.00690, 2015.

[Isa06] I. Martin Isaacs. Character Theory of Finite Groups. AMS Chelsea PublishingSeries. AMS Chelsea Pub., 2006.



[KI03] Valentine Kabanets and Russell Impagliazzo. Derandomizing Polynomial Iden-tity Tests Means Proving Circuit Lower Bounds. pages 355–364, 2003.

[Kin94] Alastair King. Moduli of Representations of Finite Dimensional Algebras. TheQuarterly Journal of Mathematics, 45(4):515–530, Dec 1994.

[Lam91] T.Y. Lam. A First Course in Non-commutative Rings. Springer-Verlag NewYork, 1991. ISBN 9780387953250.

[Lan12] Joseph M. Landsberg. Tensors: Geometry and Applications, volume 128 ofGraduate Studies in Mathematics. American Mathematical Society, Provi-dence, RI, 2012.

[LP90] Lieven Le Bruyn and Claudio Procesi. Semisimple Representations of Quivers.Transactions of American Mathematical Society, 317, Feb 1990.

[MFK94] David Mumford, John Fogarty, and Francis Kirwan. Geometric InvariantTheory. Springer-Verlag Berlin Heidelberg, 1994. ISBN 9783540569633.

[Pro76] Claudio Procesi. The invariant theory of n× n matrices. Advances in Mathe-matics, 19(3):306 – 381, Mar 1976.

[Raz74] Yurii P. Razmyslov. Identities With Trace in Full Matrix Algebras Over a Fieldof Characteristic Zero. Izv. Akad. Nauk SSSR Ser. Mat., 38:723–756, 1974.

[Stu93] Bernd Sturmfels. Algorithms in Invariant Theory. Springer-Verlag, Berlin,Heidelberg, 1993.

[SV99] Aidan Schofield and Michel Van den Bergh. Semi-Invariants of Quivers for Ar-bitrary Dimension Vectors. arXiv Mathematics e-prints, page math/9907174,Jul 1999.

[TC38] Robert M. Thrall and Josephine H. Chanler. Ternary Trilinear Forms in theField of Complex Numbers. Duke Math. J., 4(4):678–690, 12 1938.

[Val79] L. G. Valiant. Completeness Classes in Algebra. pages 249–261, 1979.

[Wes00] Douglas B. West. Introduction to Graph Theory. Prentice Hall, 2 edition,September 2000.

[Wey66] Hermann Weyl. The Classical Groups: Their Invariants and Representations.Princeton University Press, 1966.


technische universit¨at berlin...matrix invariants and semi-invariants we are also able to give...

Documents