computing with multiplicative invariantslorenz/talks/compmit.pdf · classical invariant theory: a...

Computing with Multiplicative Invariants

Euler Symposium at Temple U 04-16-2007

Martin Lorenz

Temple University

Philadelphia

Euler Symposium 04-16-2007 – p. 1/21

http://math.temple.edu/~lorenz/

Apology

Leonhard Euler1707 - 1783


http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Euler.html

Overview

Classical invariant theory: a brief introduction


Overview


Multiplicative invariants: definition and somecomputational issues


Overview



Regularity: the significance of reflections


Overview



Regularity: the significance of reflections

The Cohen-Macaulay property: exploring the unknown


Part I: Classical invariant theory


Algebraic formulation

Given: a linear group

G ⊆ GLn(k)

over some field k (usually C)




G ⊆ GLn(k)


G acts on the polynomial algebra

k[x1, . . . , xn]

by "linear substitutions of the variables"




G ⊆ GLn(k)


G acts on the polynomial algebra

k[x1, . . . , xn]

by "linear substitutions of the variables"

The algebra of (polynomial) invariants is

k[x1, . . . , xn]G = {f ∈ k[x1, . . . , xn] | g(f) = f ∀g ∈ G}


Geometric view

k[x1, . . . , xn] oo //_______ affine spacekn

��

k[x1, . . . , xn]G oo //________

� ?

OO

quotientkn/G


Example #1

Linear group: G ={(

1 00 1

)

, g =(

−1 00 −1

)}

action on k[x, y]: g(x) = −x, g(y) = −y

⇓

invariants:

only even degrees:

k[x, y]G = k[x2, y2, xy]


Example #1

geometric view:

k[x, y] oo //___________ affine plane k2

��

k[x, y]G = k[x2, y2, xy] oo //________

� ?

OO


Pioneers of invariant theory

Masters of computation:

Aronhold (1819 - 1884)

Clebsch (1833 - 1872)

Gordan (1837 - 1912)

Cayley (1821 - 1895)

Sylvester (1814 - 1897)

Cremona (1830 - 1903)

. . .


Pioneers of invariant theory

Abstract approach:

David Hilbert1862 - 1943


http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hilbert.html

Part II: Multiplicative Invariants


The setting

Given: a group G and a G-lattice L ∼= Zn; so

G→GL(L) ∼= GLn(Z)

an integral representation of G


The setting



Choose a base ring k and form the group algebra

k[L] =⊕

m∈L

kxm ∼= k[x±1

1 , . . . , x±1n ] , x

mx

m′

= xm+m′

The G-action on L extends uniquely to a “multiplicative”action by k-algebra automorphisms on k[L]:

g(xm) = xg(m) (g ∈ G,m ∈ L)


The setting



Choose a base ring k and form the group algebra

k[L] =⊕

m∈L

kxm ∼= k[x±1

1 , . . . , x±1n ] , x

mx

m′

= xm+m′

The G-action on L extends uniquely to a “multiplicative”action by k-algebra automorphisms on k[L] .

The multiplicative invariant algebra is

k[L]G = {f ∈ k[L] | g(f) = f ∀g ∈ G}


Example #2

Multiplicative inversion in rank 2: G = 〈g | g2 = 1〉

L = Ze1 ⊕ Ze2

action: g(ei) = −ei


Example #2


L = Ze1 ⊕ Ze2


Putting xi = xei we have:

k[L] = k[x±11 , x±1

2 ] with g(xi) = x−1i


Example #2


L = Ze1 ⊕ Ze2


Straightforward calculation gives

k[L]G = k[ξ1, ξ2] ⊕ ηk[ξ1, ξ2]

with ξi = xi + x−1i and η = x1x2 + x−1

1 x−12 ; they satisfy

ηξ1ξ2 = η2 + ξ21 + ξ2

2 − 4


Example #2


L = Ze1 ⊕ Ze2


Hence: k[L]G ∼= k[x, y, z]/(x2 + y2 + z2 − xyz − 4)


Some computational issues

Back to general multiplicative actions:

G a finite groupL a G-latticek a commutative base ringk[L] the group algebra



In general, k[L] has no grading ( with only k in degree 0)that is preserved by the action of G

computational theory not yet highly developed∃ some GAP & MAGMA-programs (L., Marc Renault)

But . . .



Jordan (1880): GLn(Z) has only finitely many finite subgroupsup to conjugacy

there are only finitely many multiplicative invariantalgebras k[L]G (up to ∼=) with rank L bounded



Jordan (1880): GLn(Z) has only finitely many finite subgroupsup to conjugacy

Minkowski (1887): The least common multiple of their ordersis given by

Mn =∏

p

p⌊ n

p−1⌋+j

n

p(p−1)

k

+j

n

p2(p−1)

k

+...



n # fin. G ≤ GLn(Z) # max’l G Mn

(up to conj.) (up to conj.)

1 2 1 2

2 13 2 24

3 73 4 48

4 710 9 5760

5 6079 17 11520

6 85311 39 2903040



Accessing these groups via computer:

All finite subgroups G ≤ GLn(Z) with n ≤ 4 areavailable through GAP

GAP and MAGMA both have data bases of allmaximal finite subgroups of GLn(Z) for n ≤ 31

The specialized computer algebra system CARATprovides all finite G ≤ GLn(Z) (and more) up to n = 6


http://www.gap-system.org


http://magma.maths.usyd.edu.au/magma/

http://wwwb.math.rwth-aachen.de/carat/

Part III: Regularity


Regularity and reflections

Notations: G a finite groupL ∼= Zn a faithful G-latticek = k a field with char k ∤ |G|

Will explain the following result . . .



Theorem 1 TFAE (1) k[L]G is regular(2)G acts as a reflection group on L

and H1(G/D, LD) = 0

(3) k[L]G ∼= k[Zr+ ⊕ Zs]

(4) ∃ root system Φ s.t. L/LG ∼= Λ(Φ)

and G = W(Φ)

Here, D is the subgroup of G that is generated by the“diagonalizable” reflections, conjugate in GL(L) to

d =

(

−11

...1

)



A search of the crystallographic groups libraryin GAP yields

n# finite G ≤ GLn(Z)

(up to conjugacy)# reflection groups G

(up to conjugacy)# cases withk[L]G regular

2 13 9 7

3 73 29 18

4 710 102 51



Example #3: the root latticeAn−1

Notation: Un =⊕n

1 Zei∼= Zn

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Zn−1

Sn-action: σ(ei) = eσ(i) (σ ∈ Sn)

Note: Sn acts as a reflection group;transpositions are reflections


Example #3: the root latticeAn−1

Notation: Un =⊕n

1 Zei∼= Zn

An−1 = {∑

i ziei ∈ Un |∑

i zi = 0} ∼= Zn−1

Sn-action: σ(ei) = eσ(i) (σ ∈ Sn)

C[An−1]Sn is not regular:

(n > 2; picture for n = 3)


Part IV: The Cohen-Macaulay Property


CM Rings

Hypotheses: R a comm. noetherian ringa an ideal of R


CM Rings


Always:

height a ≥ depth a = inf{i | H ia(R) 6= 0}


CM Rings


Always:

height a ≥ depth a = inf{i | H ia(R) 6= 0}

(Zariski) topologydimension theory

(homological) algebra

Def: R is Cohen-Macaulay iff equalityholds for all (maximal) ideals a


Some Examples of CM Rings

Standard example: R an affine domain/PID k, finite /some polynomial subalgebra P = k[x1, . . . , xn]. Then:

R CM ⇔ R is free over P


Some Examples of CM Rings

Standard example: R an affine domain/PID k, finite /some polynomial subalgebra P = k[x1, . . . , xn]. Then:

R CM ⇔ R is free over P

Hierarchy: catenary

regular +3 completeT +3 Gorenstein +3 CM

KS

dim 0

5=rrrrrrrrrrr

rrrrrrrrrrr dim 1reduced

KS

dim 2normal

`h HHHHHHHH

HHHHHHHH


Multiplicative Invariants: CM-property

Notations: G is a finite group 6= 1

L a G-lattice, WLOG faithful





If |G|−1 ∈ k ("non-modular case") then k[L]G is certainly CM;otherwise usually not

Will concentrate on k = Z





Theorem 2(L, TAMS ’06)

If Z[L]G is CM then all Gm/R2(Gm) for m ∈ L

are perfect groups, but not all Gm are.

subgroup gen. bybi-reflections on L:rank(gL − IdL) ≤ 2





Theorem 2(L, TAMS ’06)

If Z[L]G is CM then all Gm/R2(Gm) for m ∈ L

are perfect groups, but not all Gm are.

Corollary (“3-copies conjecture”) Z[L⊕r]G is never CMfor r ≥ 3.


Test case: the groupSn

Problem What are the Sn-lattices L

such that Z[L]Sn is CM ?



Problem What are the Sn-lattices L

such that Z[L]Sn is CM ?

Theorem 2 and classification results for certain finite lineargroups (Huffman and Wales, 70s) allow to reduce this problem to thefollowing question about polynomial invariants . . .



Problem’(open for p ≤ n/2)

Are the "vector invariants"Fp[x1, . . . , xn, y1, . . . , yn]Sn CM?





1st open case n = 4, p = 2 is ok! (Thanks to MAGMA)

> n:=4; p:=2;

> G:=PermutationGroup< 2*n | (i,i+1)(n+i,n+i+1) : i in [1..n-1] >;

> R:=InvariantRing(G,GF(p));

> IsCohenMacaulay(R);

true







It follows that Problem’ is ok for n ≤ 5 (easy argument)







It follows that Problem’ is ok for n ≤ 5 (easy argument)

Next open cases n = 6, p = 2 and p = 3: out of memory!



Summary

Let L be a G-lattice, where G is a finite group.

G is generated byreflections on L

Bourbaki, Farkas

L.*2 Z[L]G is asemigroup algebra

?

jr

Hochster

��

G is generated bybireflections on L

Z[L]G isCohen-Macaulay

?

ks


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