notes 6: representations — basics.€¦ · notes 6: representations — basics. version 1.00 —...

Notes 6: Representations — basics.

Version 1.00 — still with misprints, but hopefully fewer. More seriously, at the endI have added a paragraph on the representations of a direct product.

Basic concepts. In this section G will as usual be a Lie group. We are mostlygoing to study representations of compact groups. This includes of course finitegroups as a very important subclass. Many of the results we shall present concernonly the finite groups, and they will by necessity be treated separately. Howeverthe results are common for the finite and the infinit case will of course be treatedcommonly, often with a remark on the finite case.

The representation theory over the complex nubers is easier than over the realsdue to the fact that all operators have eigenvalues in the complex field. We willtherefore basically develop the theory for complex representations, but always withan eye on the real case.

Vector space will always be over K which as usual is either R or C (and occatio-nally it can be the division algebra H). We shall with a few exceptions assume thatall vector spaces are finite dimensional.

By a representation of G on (a finite dimensional) vector space V over K, wemean a Lie group homomorphism

ρ : G → Gl(V ).

that is a continuous group homomorphism. If G is finite (or more generally discrete)the assumption that ρ be continuous is superfluous.

As shorthand for ρ(g)(v) we write gv. The the condition that ρ be a homorphismtranslates into the two relations g(g�v) = (gg

�)v and ev = v, which must be valid forall v ∈ V and all g, g

� ∈ G.The representations of G form a category whose maps are thew K-linear maps

respecting the action of G, that is, if V and W denotes two representations of G, aK-linear map φ : V → W satisfying φ(gv) = gφ(v) for all g ∈ G and all v ∈ V . Theset of such maps is denoted by HomG(V, W ). There is an obvious inclusion of vectorspaces

HomG(V, W )⊆ HomK(V, W )

with HomG(V, W ) being the subset of maps commuting with the action of G.

Notes 6: Repteori — basics MAT4270 — Høst 2012

There is a flourishing terminology when it comes to these maps: They are calledG-equivariant, G-maps or G-module homomorphisms or some times also intertwiningoperators. I’ll probably use an equidistibuted mix of those names.

We let RepKG denote the category of representations of G of finite dimension. It

is a very nice category with a lot of structure. A s we will soon see, it is an abeliancategory with kernels, cokernels, exact sequences etc.

Invariants. We are escpecially interested in the invariants of an action. That isall the vectors v ∈ V not affected by G,i.e., thos satisfying gv = v for g ∈ G. Theyform a linear subspace wich we denote by V

G or sometimes by ΓGV , so

VG = ΓGV = { v ∈ V | gv = v for all g ∈ G }.

Clearly a G-equivariant map takes invariants into invariants, hence ΓG is a functor

ΓG : RepKG → VectSpcs

K.

This functor is additive and left exact, but not exact in general.

Example �. — The functor ΓG is not exact. One of the simplest example ofthis phenomenon is G = R acting on V = R

2 via the matrix

ρ(t) =

�1 t

0 1

�.

One easily check that ΓGV = Re1, where e1 denotes the basis vector (1, 0)t, andthat there is an exact sequence

0 �� Re1�� V

ψ�� V/Re1

�� 0

where the quotien V/Re1 � R is equiped with the trivial action. Hence ΓGψ = 0.❅

Invariant subspaces and quotients. A subspace W⊆V of a representation issaid to be invariant if gW⊆W for all g ∈ G. If W⊆V is invariant, the action of G

passes to the quotient V/W ; indeed, an element g ∈ G acts on V/W by sending acoset v + W to the coset gv + W , and this is well defined since W is invariant. Oneeasily verifies the axioms for an action.

This shows that the category RepKG has kernels and cokernels (and that the

kernel of the cokernel is equal to the cokernel of the kernel!).

— 2 —


The trivial representations. A representation of G on V is trivial if gv = v

for all g and all v. The one-dimensional trivial representation will sometimes bedenoted by G, or some times with K if there is no risk of it being confused withother one-dimensional representations.

Trivially, any vector space has a trivial representation. A fancy way of expressingthis is to say that there is an embedding of categories VectK⊆Rep

KG.

Direct sums. If V and W are two representations of G, then we equip the directsum V ⊕W with the “parallelle” action g(v, w) = (gv, gw). It is the direct sum inthe category Rep

KG.

In general, if V ��

iVi, some of the Vi might be isomorphic representations,

and we often group those together and write V ��

iniVi where niVi stands for the

direct sum Vi ⊕ Vi ⊕ · · ·⊕ Vi of ni copies of Vi.

Internal hom. The vector space HomK(V, W ) has a natural action of G. If φ ∈HomK(V, W ) and g ∈ G, we let g act on φ by g

φ = g ◦ φ ◦ φ−1. Admittedly the

notation gφ is not very good, but the action of g is a left action, and therefore g

is natural situated is somewhere to the left, and the upper left corner is not veryfrequenly in use for notation.

The following lemma is fundamental, stating that the G-maps from V to W arethe just invariants in HomK(V, W )G:Lemma 1

HomK(V, W )G = HomG(V, W )

Proof: The action on HomK(V, W ) is

gφ = g ◦ φ ◦ g

−1,

so gφ = φ if and only if g ◦ φ = φ ◦ g. ❏

There is special name for the representation V∗ = HomK(V, G). It is called the

dual or the contragredient representation. The action on it is described by gφ(v) =

φ(g−1).There is also the usefull relation:

HomK( G, V ) = VG.

Tensor products. Recall that the tensor product V ⊗K W of two vector spaces V

and W over K, is characterised by a universal property which we will now describe.

— 3 —


The bilinear map V ×W → V ⊗K W sending (v, w) to v⊗w ha the property thatall other bilinear maps β : V ×W → U may be factored

V ×Wβ

��

��

U

V ⊗K W

β�

��

,

where β� : V ⊗K W → U is a K-linear map. The map β

� is unique.Elements in V ⊗K W are sums

�vi⊗wj, and if {ei}i∈I and {fj}j∈J are basises

for V and W respectively, the elements {ei⊗ fj}(i,j)∈I×J form a basis for the tensorproduct. In parrticular, dimK V ⊗K W = dim V dim W .

To define a linear map ψ from V ⊗K W it suffices, by the universal property, tospecify ψ(v⊗w), provided this specification is is bilinear in v and w.

Let now V and W be two representations of G. One may define an action of G onthe tensor product V ⊗K W by letting g(v⊗w) = g(v)⊗ g(w) and then extend bylinearity. This makes the tensor product in to a representation, and the constructionis functorial in both V and W .

The trivial one-dimensional representation G is a neutral element for the tensorproduct: V ⊗K G = V .

There is a natural K-linear map ψ : V∗⊗W → HomK(V, W ). Indeed, given an

element φ⊗w with φ ∈ V∗ and w ∈ W , we define the ψ on decomposable tensors as

ψ(v) = φ(v)w and then extend it by linearity. In this way we obtain a G-equivariantmap since for g ∈ G, the element g

φ⊗ gw corresponds to the map v �→ gφ(v)gw =

g(φ(g−1v)w), which is just g

ψ(v).

Lemma 2 If V and W are two representations of G, then the map

V∗⊗W → HomK(V, W )

sending φ⊗w to the map v �→ φ(v)wis an isomorphism of representations.

Proof: This is a question in linear algebra, and we may forget the structures as G-modules. Both sides are additive in W , and therefore it suffices to prove the lemmafor W = K. But then it is obvious. ❏

For any natural number n, we let V⊗ n stand for V ⊗ V ⊗ . . .⊗ V where there are

n factors in the tensor product. The tensor algebra is the direct sum T (V ) =

— 4 —


�n≥0 V

⊗ n. It is an associative algebra with the product defined on decomposab-le tensors by (v1⊗ . . .⊗ vn)(w1⊗ . . .⊗ vm) = v1⊗ . . .⊗ vn⊗w1⊗ . . .⊗ vm.

Example �. — The cyclic groups µn. For any p ∈ Z the cyclic group of n-roots of unity µn = { � ∈ C | �

n = 1 } acts on C by � · z = �pz . We denote this

one-dimensional complex representation by Lp. Clearly Lp = Lp� if p ≡ p� mod n.

Moreover Lp+q � Lp⊗ Lq. This follows since there is the isomorphism of vectorspaces C⊗C C � C with 1⊗ 1 corresponding to 1. Now � acts on 1⊗ 1 as �

p ·1⊗ �p� ·

1 = �p+p

�1⊗ 1. One also has L

∗p

= L−p. ❅

Example �. — The circle S1. In complete analogy with the preceding example,

there is for each n ∈ Z a one-dimensional complex representation Ln of the circleS

1 given by z ·w = znw. As above Ln⊗ Lm = Ln+m and L

∗n

= L−n, but in this caseLn �� Lm for n �= m. ❅

Example �. — The symmetric powers. Let V be a vector space and letx1, . . . , xm be a basis for V . We have the symmetric algebra Sym(V) =

�n≥0 Symn(V),

where Symn(V) is the subspace of V⊗ n consisting of symmetric tensors. That is, ten-

sors�

vi1 ⊗ . . .⊗ vinsuch that

�viσ(1)

⊗ . . .⊗ viσ(n)=

�vi1 ⊗ . . .⊗ vin

for any per-mutation σ. One may think about the symmetric product as the ring of polynomialsin the variables x1, . . . , xm and coefficients in K. The homogeneous polynomials ofdegree n form the vector space Symn(V).

The general linear group acts on V , hence on each of the symmetric tensorproducts Symi(V). These actions are just what we know from earlier as the “linearchanges of variables” in the polynomials. ❅

The ring of virtual representations. There is a quite general constructionwhich we in our context would call the ring of virtual representations. Informal-ly speaking, the virtual representations are finite, linear combinations of the form�

nV [V ] where nV ∈ Z and where [V ] denotes the isomorphism class of V . If all thecoefficients nV are positive, such a sum corresponds to an existing representationfrom the real world, but some av being negative, it is only virtual. There is also arelation between the formal sum [V ] + [W ] and the “real” sum V ⊕ W . And themultiplicative structure comes from the tensor product.

There are two variants of the formal construction, giving different rings in general.We shall sketch both. One starts, in both cases, with the free abelian group A onthe set of isomorphism classes of representations; such an animal being nothing buta finite, formal sum

�ini[Vi] with each ni ∈ Z.

— 5 —


Inside A we let B be the subgroup generated by all the formal sums [V ⊕W ]−[V ]− [W ], where V and W are any representations. Then the representation ring wewant is RKG := A/B. It is by definition an abelian group, and one checks readilythat the tensor product induces an associative an commutative product on RKG,distributive over the addition. We’ll write [V ] · [W ] = [V ⊗W ].

The other way of doing this, gives an a priori different ring. However, the tworings are strongly related, and in the case of compact groups, that we are interestedin, they are the same. As before, A is the free abelian group on all isomorphismclasses of finite dimensional representations of G, but the subgroup B is slightlydifferent. In this case the generatores are all the elements of the form [V ]−[V �]−[V ��]where V ,V � and V

�� fit into some exact sequence of representations

0 ��V� �� V ��

V�� 0.

Change of base field. If V is a real representation of G, then of course VC =V ⊗R C is a complex one, and this change of base field is compatible with the for-mation of direct sums and tensor products. Hence it induces a map of rings

rC : RRG → RCG.

The other way around, any complex representaion is equally a real one — just forgetthat there is a complex structure. Clearly this forgetfullness respects the direct sums,and we get a homomorphism of abelian groups

rR : RCG → RRG.

This is not a multiplicative map, e.g., C⊗C C = C, but C⊗R C is something elsebeing of real dimension 4 (se example 6 below). However, there is what one calls aprojection formula:

rR(rCV · W ) = V · rR

W.

Example �. — S1; the complex case. The complex representation rings of S

1

is RCS1 = Z[l, l−1] where l represent the isomorphism clss og L1, which follows since

Ln � L⊗ n

1 . ❅

Example �. — S1; the real case. The real ring RRS

1 is slighly more complicated,and in fact a little intricate, so it is worthwile spending some time on understandingit. It also illustrates the contrasting complexity of the real representations and thecomplex ones.

— 6 —


We start with the statement

rCC = C⊕ C.

This cryptic statement means the following: First of all, recall that any complexvector space has a concubine, the conjugate space V̄ , where the multiplication isdefined by z · v := z̄v. Now, there are two structures as C vector space on C⊗R C,and they are different. One is induced from multiplication by z⊗ 1 — that is multi-plication by z from the left — and one is induced by multiplication from the right,i.e., by 1⊗ z. And the one on rCC, is by definition the one from the right.

We have the following real basis for C⊗R C:

e1 =1

2(1⊗ 1 + i⊗ i) f1 =

1

2(i⊗ 1− 1⊗ i)

e2 =1

2(1⊗ 1− i⊗ i) f2 =

1

2(i⊗ 1 + 1⊗ i)

One verifies that any v ∈ C = Re2⊕Rf2 is such that z⊗ 1 · v = v · 1⊗ z (it sufficesto verify it for z = i), but for v ∈ C̄ = Re1⊕Rf1 left and right multiplication differ,i.e., v · 1⊗ z = z̄⊗ 1 · v.

Let us now come back to ring RRS1. First of all Ln � L−n. This follows from the

matrix equation �a −b

b a

�=

�0 11 0

� �a b

−b a

� �0 11 0

�

used with a = cos nt and b = sin nt. What about the product Ln⊗R Lm? Its un-derlying vector space is C⊗R C, with S

1 acting with eint from the left and by e

imt

from the right. In the decomposition C⊗R C = C⊕C the action of eint from the left

induces multiplication with eint on both factors, contrasting the action of e

imt fromthe right, which acts as e

imt on the first factor and as e−imt on the second factor C̄.

This shows that lnlm = ln+m + ln−m if n �= m and lnln = l2n + 2, where we writeln for the isomorphism class [Ln]. So the ring is really kind of intricate, but no worsethan the formula reflects the elementary formula

4 cos nt cos mt = 2 cos(n + m)t + 2 cos(n−m)t

which in the old days of slide rules was called the logaritmic expression for the sumof two cosines.

— 7 —


About the two maps rC and rR one can say

rRln =ln

rCln =ln ⊕ l−n,

the first equallity holds by definition, and the second one, one may concider as ajazzed up version of the formula 2 cos nt = e

int + e−int. ❅

Irreducible representations We say that a represenation is irreducible when ithas no proper, nontrivial invariant subspace; that is if W⊆V is invariant under theaction of G, i.e., gW⊆W for all g, then either W = 0 or W = V .

The irreducible representations play a crucial role in the theory of representa-tions. They are the building blocks for all other representations, and at least forthe so called semi simple groups, including compact groups, any finite dimensionalrepresentation is a direct sum of irreducible ones, as we’ll soon see.

We start with the following simple, but fundamental lemma, called Schur’s lem-ma after one of the fathers of representation theory, the German mathematicianIsaac Schur. It can be formulated in great generality and over any ground field, butwe stick with Lie groups and the fields R or C, since that is what concerns us.

Lemma 3 (Schur) Assume that G is a Lie group and that V and W are twoirreducible representations of G over K. Assume further that φ : V → V is a G-map.Then either φ is 0, or φ is an isomorphism.

Proof: Assume hat φ is not equal to the zero map. Then Ker φ = 0 since it is notthe whole of V , so φ is injective. Neither is he image Im φ zero, so Im φ = V and φ

is surjective. ❏

Corollary 1 If V is irreducible, then EndG(V ) is a division algebra over K.

Corollary 2 Assume that V is an irreducible G-representation and that φ : V → V

is a G-map. If the ground field K is algebraically closed (that is K = C), then φ ismultiplication by a scalar.

Proof: Since the ground field is algebraically closed, the characteristic polynomialof φ has a root, meaning that φ has an eigenvalue λ. Then the map λ idV −φ, which is

— 8 —


G-equivariant, has a non-trivial kernel, and V being irreducible, λ idV −φ is identicalzero. ❏

Example �. If K = R all three division algebras R, C and H occure as the endomorp-hism algebra of an irreducible representation. Examples of the two first are easi tofind, but we give one V with EndR(V ) = H. The group will be finite. It is one of theso called the quaternion groups and it is often denoted by Q8. Pick three orthogonal,pure imaginary quaternions e1, e2 and e3 = e1e2, and let Q8 = {±1,±e1,±e2,±e3}.Since the ei’s anti-commute and are of square minus one, this is a multiplicativesubgroup of H. It acts from the right on H, that is g · q := qg

−1, and since theelements of Q8 are algebra-generators for H, the G-module H is irreducible. Indeed,if V⊆H is an invariant subspace, HV⊆V , and hence V = H, since any non-zerov ∈ V is invertible.

Now, the action of H on itself by left muliplication commutes with the rightaction of G. Hence H⊆ HomQ8(H, H), and equality follows since EndH(H) = H. ❅

Example �. — The commutative case. An instantaneously application ofSchur’s lemma is to the commutative groups and the following observation, whichis just another version of the principle that commuting elements have common ei-genvectors. First, some more terminology. A complex, multiplicative character of theLie group G is a Lie group homomorphism χ : G → C

∗.Examples would be the determinant det : Gl(n, C) → C

∗ or if Tn = S

1× . . .×S1

is a torus, any of the projections onto one of the factors followed by a multiplicativecharacter of S

1 into C∗. We have seen that Hom(S1

, S1) = Z, so the multiplicative

characters of S1 are just the maps z �→ z

n.

Corollary 3 If G is commutative, any irreducible, complex representation of G isone-dimensional, and it is given by a complex, multiplicative character.

Proof: Let g ∈ G. Then the action v �→ gv commutes with the action of any otherelement in G. Hence, after Schur’s lemma, there is complex number χ(g) such thatg ·v = χ(g)v. This is true for all g, so any nonzero vector spans an invariant subspace.Since V irreducible, V = � v � and dim V = 1.

Now Gl(1, C) = C∗, so the representation is just a multiplicative map G → C

∗,that is a multiplicative character. ❏

❅

— 9 —


Complete reducibility The following theorem is one of the fundamental resultsin the representation theory of compact groups. It is really the starting point forthe theory. To prove it we need some machinery to be able to take averages offunctions over the group. These averages will be invariant under the group action —like a center of mass must be invariant under all symmetries of the system — so insome sense, they furnish us with a systematic way of generating invariants. Assumef : G → V is a smooth map with values in some representation V . If G is finite,such an average would be |G|−1

�g∈G

f(g), while in case of G being infinit, we haveto replace the sum by a suitable integral, called the Haar-integral. We’ll soon comeback to that, but first we state the result:

Theorem 1 If G is a compact Lie group, and V is a finite dimensional representa-tion over K, then V splits as a direct sum of irreducible representations.

This result is often stated as V being completely reducible. In case G is finite,the same statement is true over any field in which the order of G is invertible. Thatresult is called Maschkes theorem. Recall that a complex representation V is calledunitary if there is a hermitian inner product � v, w � on V invariant under G; i.e.,� gv, gw � = � v, w � for all g ∈ G and all v, w ∈ V . The proof of the theorem has twoparts, the first being:

Proposition 1 If V is a unitary representation of G, then V splits as a direct sumof irreducible representations.

Proof: The proof goes by induction on the dimension of V , the case dim V = 1posing no problem. Assume that V is not irreducible. Then we may find a proper,invariant and non-trivial subspace W⊆V . Since the representation is unitary, thereis an invariant hermitian form on V . Let W

⊥ be the orthogonal complement to W .Then W

⊥ is invariant; indeed, if w ∈ W⊥ and v ∈ W we have

� gw, v � =�w, g

−1v

�= 0.

Then V = W⊕W⊥ is a decomposition of V into two proper, invariant subspaces, and

by induction, each one of them can be written as a sum of irreducible representations.❏

Let us remark that in the real case, the same argument goes through , if weassume that the representation is othogonal, i.e., has a positive definit invariantform.

The second part of the proof of the theorem is the following, the proof of whichis where the averaging is required:

— 10 —


Proposition 2 Assume that G is a compact Lie group and that V is a finite di-mensional complex representation of G. Then there is an invariant hermitian formon V . In other words, the representation is unitary.

Proof: In the case the group G is finite, we can prove this straight away: Pick anyhermitian form on V and denoted it by (v, w). Define a new hermitian form, � v, w �,by averaging the old one over G:

� v, w � = |G|−1�

g∈G

(gv, gw).

First of all, the new form � v, w � is invariant:

�hv, hw � = |G|−1�

g∈G

(ghv, ghw) = |G|−1�

g∈G

(gv, gw) = � v, w �

since gh runs through G when g does. The new form inherits symmetry and bilinea-rity from the old one, so the only thing left to verify is the non-degeneracy. But ifv �= 0, then

� v, v � = |G|−1�

g∈G

(gv, gv) > 0

since each gv �= 0, and the old form (v, w) is definite. ❏

Example �. — The complex abelian case. Following up our example aboutabelian groups from the last section, we can now say that if G is compact andcommutative, then any finite dimensional complex representation of G is a directsum of one-dimensional ones. ❅

Example ��. — The real abelian case. The real irreducible representationsof S

1 are not one -dimensional, most of then being of dimension two. Every complexrepresentation Ln is of course a real one if we forget the complex structure. Werecognise the action of z = e

it by multiplication by zn as the rotation of R

2 an anglent. With the exception of n = 0, which is the identity representation S1 , they areall irreducible; any line through the origin is moved.

W’ll verify that the representations Ln are all the irreducible ones.Indeed, if V is a real irreducible S

1-module, we may regard the complex repre-sentation VC = V ⊗R C. It has a complex conjugation map w �→ w̄ defined byv⊗ z �→ v⊗ z̄, so we we can recognize V inside VC as V = V ⊗R R⊆V ⊗R C = VC ,

— 11 —


i.e., the subset of all vectors with w̄ = w. We may think about V as the real partof VC.

The complex representation VC decomposes into a sum of complex one-dimensionalones. Let L be one of those. Then L has a generator v, that is an eigenvalue for allz ∈ S

1 associated to some multiplicative character λ. As the representation is real, v̄

is also an eigenvector, and its character is λ̄. Hence e1 = v + v̄ and e2 = i(v− v̄) arereal vectors and, if λ �= λ̄, they generate a real two dimensional subspace of V . Onechecks that ze1 = Re λ(z)e1 + Im λ(z)e2 and ze2 = − Im λ(z)e1 + Re λ(z)e2. Hencethe span � e1, e2 � is of the of the requiered type.

If λ = λ̄, then λ is a real character of S1i.e., a continuos group homomorphism

map S1 → R

∗. The image is a compact, connected subgroup, hence the trivialsubgroup. ❅

The isotypical components Let W be any (finite dimensional) G-module. Wemay write W =

�iVi with each of th Vi irreducible. This decomposition is not

unique, e.g., if two or more of the Vi are ismorphich, but grouping the isomorphicirreducible components together, we get a canonical decompostion of W . The cleanway of doing this is as follows. We start with defining the map dV for any irreducibleV :

dV : HomG(V, W )⊗V → W given by φ⊗w �→ φ(w).

Here V can be any irreducible, but if V does not occure as a summand in W ,certainly HomG(V, W ) = 0.

Summing up over V , we get a map

IW :�

V irreducible

HomG(V, W )⊗V → W

.

Proposition 3 The map IW is a G-module isomorphism.

Proof: The map IW is additive in W , so it suffices to check the proposition whenW is irreducible. But then HomG(V, W ) = 0 if W �= V and HomG(W,W ) = K · idW ,so IW reduces to dW : K⊗W → W sending α · idW ⊗w to αw. This is clearly anisomorphism. ❏

We call HomG(V, W )⊗ V the isotypical component of W and the dimensiondimK HomG(V, W ) the multiplicity of V in W .

— 12 —


The Haar integral

Let G be a Lie group. If µ is a Borel measure on G — i.e., a measure such thatalle the open sets are measurable — we say that µ is a left invariant measure ifµ(X) = µ(gX) for any measurable subset X⊆G.

It is a result that there exists such left invariant measures, called Haar measures,with the additional property that µU > 0 for any non-empty open subset, andµK < ∞ for K⊆G compact. More over such a measure is unique up to scale. Sowith our hypothesis that G be compact, we may normalise it such that µ(G) = 1,and then this makes it unique.

It follows that for any measurable functionon G the integral is G-invariant:

�

G

f(gx) dx =

�

G

f(x) dx (✮)

Two comments: First, what one can do with left invariant measures one canequally well do with right invariant ones. So there are also right invariant integrals.But in general — as in politics — the right and the left do not agree. The twointegrals may differ, even if we allow scaling. However for compact groups theycoincide, when normalised, so this left-right discrepancy is not an issue for us.

We were not precise about where the function in ✮ takes values, and infact, itcan be in any vector space over K. If e1, . . . , en is a basis for V a function f : G → V

may be written f(x) =�

ifi(x)ei where each fi is a scalar function. Then

�

G

f(g) dg =�

i

(

�

G

fi(g) dg)ei.

One easily convinces oneself that the following lemma is true, which in fact justsays that the integral is linear:

Lemma 4 If f : G → V is measurable and A : V → W is a linear map, then A ◦ f

is measureable and

A(

�

G

f(g) dg) =

�

G

A(f(g)) dg

In particular if V is a G-module, we may take for A the map v �→ hv for anyh ∈ G. Then

h

�

G

f(g) dg =

�

G

hf(g) dg.

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As an urgent application of the Haar integral, let us finish the proof of proposition2. The argument follows the same lines as in the finite case. Chose any hermitianform (v, w) on V and introduce a new one by the formula

� v, w � =

�

G

(gv, gw) dg.

Then � v, w � is an invariant form, to see this we use, in fact, the right invariance ofthe integral:

�hv, hw � =

�

G

(ghv, ghw) dg =

�

G

(gv, gw) dg = � v, w � .

As in the finite case, the properties of symmetry and linearity are inherited, andnon-degeneracy follows since if v �= 0,

� v, v � =

�

G

(gv, gv) dg > 0

the function (gv, gv) being positive for all g ∈ G.

Proposition 4 Let the compact Lie group act on V . Then the map pV : V → V

given by

pV (v) =

�

G

gv dg

is a projection onto VG. That is p

2V

= pV and pV (v) = v if v ∈ VG. Furthermore, this

construction is functorial; that is if φ : V → W is a G-map, then φpV (v) = pW (φv).

Proof: There are three things to prove. First that pV takes values in VG:

hpV (v) = h

�

G

gv dg1=

�

G

hgv dg2=

�

G

gv dg = pV (v).

where equality 1 is by linearity of the integral (lemma 4) and number 2 holds becausethe integral is translation invariant.

The second thing to prove is that pV (v) = v if v ∈ VG, so assume thar gv = v

for all g ∈ G: �

G

gv dg =

�

G

v dg = v,

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since the integral is normalised so that�

Gdg = 1. That p

2V

= pV follows from this.The functoriality follows from lemma 4:

φpV (v) = φ

�

G

gv dg1=

�

G

φgv dg2=

�

G

gφv dg = pW φv,

where equality number 1 is linearity of the integral and number 2 hods since φ is aG-map. ❏

Proposition 5 Every exact sequence of finite dimensional G-modules

0 ��V� �� V

ψ��V�� 0

is split exact as G-modules.

Proof: We apply the averaging procedure to the G-module HomK(V ��, V ), so let

p : HomK(V ��, V ) → HomG(V ��

, V ) be the projection operator as in the previousproposition. Chose any linear section of ψ, that is an elemenet σ ∈ HomK(V ��

, V )with ψ ◦ σ = idV �� . Then p(σ) is G-map, and we claim that it still is a section of ψ,indeed ψ ◦ p(σ) = p(ψ ◦ σ) = p(id) = id by functoriality and the fact that the mapHomK(V ��

, V ) → σ ∈ HomK(V ��, V

��) sending ξ → ψ ◦ ξ is a G-map since ψ is. ❏

Corollary 4 The functor ΓG is an exact functor from RepKG to Rep

KG.

Characters and the trace. The trace plays an overly important role in thetheory of representations. We start by recalling the definition of the trace of anendomorphism φ : V → V of a finite dimensional vector space V over K.

In terms of a basis {ei} of V and the matrix (aij) of φ in that basis, the trace ofφ is the sum of the diagonal elements: tr φ =

�iaii. Another way of saying this, it

is the negative of the next highest term of the characteristic polynomial:

det(λ · idV − φ) = λn − tr φλ

n−1 + . . . .

If φ can be diagonalised, the trace is just the sum of the eigenvalues λi. Thenthe traces of the powers φ

n are the sums of the n-th powers of the eigenvalues; i.e.,tr φ

n =�

iλ

n

i.

It is a general fact — going back to Newton — that the elementary symmetricfunctions — which are the coefficients of a monic polynomial on terms of the roots

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— can be recovered from the power symmetric functions. For example, the easiestinstant of this is the formula λ1λ2 = 1/2((λ1 + λ2)2 − (λ2

1 + λ22)). So knowing tr φ

n

for all n, we know the characteristic polynomial of φ. Hence the eigenvalues, andsince we assumed that φ can be diagonalised, φ it self is determined up to conjugacy.This is the underlying fact of why the character of a representation play such animportant role:

Lemma 5 If φ and ψ are two diagonalisible endomorphisms of V and tr φn = tr ψ

n

for all n ∈ N, then φ = aψa−1 for som a ∈ Gl(V ).

This motivates the following definition:

Defenition 1 Let G be a Lie group and let V be a G-module. Then we define thecharacter of V as the map χV : G → K given by χV (g) = tr g|V , where by g|V weunderstand the map v �→ gv.

We call a function f : G → K a class function if f is constant on the conjugacyclasses of G. That is if f(gxg

−1) = f(x) for all x, g ∈ G. The characters are classfunctions since the trace is invariant under conjugacy.

The different properties of the trace translate into properties of the characters:

Proposition 6 Let V and W be finite dimensional representations of the Lie groupG. Then

1. χV⊕W = χV + χW .

2. χV ⊗W = χV χW .

3. χV ∗(g) = χV (g−1). If K is the field C of comple numbers and V is a unitaryrepresentation, then χV ∗ = χV

4. χV (e) = dimK V

Proof: Exercise. ❏

A fundamental property of the trace, which is the root of most fixed point for-mulas, is that if V is a vector space and p : V → V a projection, that is p

2 = p, thentr p = dim Im p. We formulate it is a lemma:

Lemma 6 If p : V → V is a projection, then tr p = dim Im p.

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Proof: There is splitting V = Im p ⊕ Ker p; indeed idV − p and p are orthogonalidempotens with sum equal to idV . Choose then a basis for Im V and one for Ker V .Together the make up a basis for V , and with respect to this basis, where the basiselements in Im V are not moved by p, the matrix of p looks like

1 0 0 0 0 · · · 0 00 1 0 0 0 · · · 0 0

0 0. . . 0 0 · · · 0 0

0 0 0 1 0 · · · 0 00 0 0 0 0 · · · 0 0...

......

...... · · · ...

...0 0 0 0 0 · · · 0 00 0 0 0 0 · · · 0 0

,

and the trace of p equals the number of non-zero columns in the matrix, that is thedimension of Im p. ❏

Proposition 7 Let G be a Lie group and V a finite dimensional representation ofG. Then we have

dim VG =

�

G

χ(g) dg.

Proof: By proposition 4 we know that pV (v) =�

Ggv dg is a projection onto V

G.Hence by lemma 6 (equality 1) and lemma 4 with A the linear map tr : HomK(V, V ) →K (equality 2)

dim VG 1

= tr p = tr

�

G

g dg2=

�

G

tr g|V dg =

�

G

χ(g) dg.

❏

If G is a finite group, the integral reduces to the sum over all elements in G, andwe get in that case:

dim VG =

1

|G|�

g∈G

χ(g).

Example ��. — Permutation representations og finite groups.. In thisexample G is a finite group acting on a finite set X. Let K[X] be the vector space of

— 17 —


formal linear combinations of the elements in X. The vectors in X has the appearenceas v =

�x∈X

fxx where fx ∈ K. The group G acts on K[X] as gv =�

x∈Xfxgx,

and the resulting representation is called the permutation representation induced bythe action on X.

Of course one may also think about K[X] as the vector space of functions f : X →K, with the action g

f(x) = f(g−1x).

The matrices of the g ∈ G are all permutation matrices, that is , matrices witheach column having zeros everywhere except at one place where there is a one. Theone is situated on the diagonal precisely when the corresponding basis element x isleft fixed by g. Hence

tr g = #{x ∈ X | gx = x }.

For the whole group, we get when as usual XG denote the fixed point set of G;

i.e., XG = {x ∈ X | gx = x for all g ∈ G }, that #X

G = |G|−1�

g∈GχK[X](g). ❅

Characters and maps between representations. Recall that if V and W

are two representations of G, then HomG(V, W ) = HomK(V, W )G. Furthermore weknow that as G-representations there is a canonical isomorphism HomK(V, W ) �V∗⊗W . This implies that the character ψ of HomK(V, W ) is equal to χ̄V χW , and

by proposition 4 this gives us the formula

dim HomG(V, W ) =

�

G

χV(g)χW (g) dg = �χV , χW �

where the inner product is the old well known one giving the L2-norm on the space

of functions on G. For the finite case we get

dim HomG(V, W ) =1

|G|�

g∈G

χV(g)χW (g).

Combining this with Schur’s lemma on page 8, we get

Theorem 2 Let V and W be two irreducible, complex representations of the compactLie group G. Then

�χV , χW � =

�

G

χV(g)χW (g) dg =

�0 if V and W are not isomorphic

1 if V and W are isomorphic.

— 18 —


where we need that the representations involved be complex to be sure that �χV , χV � =1. In the real case �χV , χV � = dim EndG(V ) =: mV which is either 1, 2 or 4.

A slightly different statement is the following:

Theorem 3 The characters of the finite dimensional, complex irreducible represen-tations of a compact Lie group G are orthogonal with the respect to the L

2 innerproduct.

Let W and V be representations of G and assume that V is irreducible. Recallthat the multiplicity nV of V in W is the number of times V occures in a de-composition of W as a direct sum of irreducibels. We showed the formula nV =dim HomG(V, W ) earlier, and this gives us

Proposition 8 The multiplicity nV of the irreducible G-module V in the G-moduleW is given by

nV = �χV , χW � .

Corollary 5 A complex representation V of G is irreducible if and only if �χV , χV � =1.

Proof: Indded, V =�

niVi with the Vi’s different irreducibles, and one finds bythe orthogonality in the theorem

1 = �χV , χV � =�

n2i�χVi

, χVi�

and since �χVi, χVi

� > 0 if Vi �= 0, this shows that there can only be one Vi in thesum and it must have multiplicity one. ❏

The proposition above combined with the fact that finite dimensional represen-tations of a compact group are completly reducible, results in the following:

Theorem 4 The complex, finite dimensional representations of the compact Liegroup G are up to isomorphism determined by their characters, i.e., if V and W areof the sort, then χV (g) = χW (g) for all g ∈ G implies that V � W .

Using the representation ring we introduced, the character is a C-algebra ho-morphism χ : RCG → C

∞(G; C), and we have shown that it is injective — in fact itis an isometry on its image, when C

∞(G; C) is equiped with the L2-inner product.

— 19 —


The direct product of groups. It is of course important to understand therelation between the representatons of a direct product of two Lie groups and theones of of the two factors. There is a very simple and natural relation, the irreduciblerepresentations of G × H being the tensor product of one irreducibel from G andone from H. To be precise, we need to introduce some notation:

Functoriality Let G and H be two Lie groups, and let φ : G → H be a Lie grouphomomorphism. Any representation ρ : H → Gl(V ) of H on V induces a “pulledback” representation φ

∗V of G on V simply being the composition ρ◦φ : G → Gl(V ).

If G⊆H is a subgroup, this is mostly called the restriction of V to G in the litteratureand denoted by redH

GV .

Since φ∗ do not change the linear maps between repesentations of H, this induces

an exact additive functor commuting with tensor products

φ∗ : RepKH → RepKG,

and hence a ring homomorphism between the two representation rings

φ∗ : RKH → RKG.

We defined two variants of the representation rings — coinciding in our case ofcompact groups — and in both cases we get a pull back map. As always, the associa-tion φ �→ φ

∗ is functorial in a contravariant way, meaning that (ψφ)∗ = φ∗ψ∗; indeed

this is just another way of expressing that composition is associative: ρ ◦ (ψ ◦ φ) =(ρ ◦ ψ) ◦ φ.

The case of a direct product Coming back to the product G×H, the mostnatural way of constructing representation of the product from representations of G

and H respectively, is what is often called the external tensor product. That is, thetensor product of two representations being pull backs from the two factors, and ,as we’ll see, this is basically all that happens, at least in the case of compact groups.To be precise, let V and W be representations of G and H respectively. Then theexternal tensor product is π

∗GV ⊗ π

∗H

W . It is often denoted by V � W .On the level of representation rings this appears as follows. The pull backs along

the two projections πg and πM induce a ring homomorphism

RKG⊗Z

RKH → RKG×H,

by sending [V ]⊗ [W ] to π∗G[V ] · π∗

H[W ] = [V � W ].

— 20 —


Proposition 9 If V and W are complex, irreducible representations of compact Liegroups G and H respectively, then V � W is an irreducible representation of G×H

We get the following corollary since the isomorphism classes of the irreduciblesform a linear Z-basis of the representation rings:

Corollary 6 The natural ring homomorphism

RCG⊗Z

RCH → RC(G×H),

is an isomorphism

Proof of the proposition.: We appeal to corollary 5 and use Fubini’ theoremon double integrals:

�χ[V �W ], χ[V �W ]

�=

�

G×H

χV(g)χ

W(h)χV (g)χW (h) dg dh =

=

�

G

χV(g)χV (g) dg ·

�

H

χW

(h)χW (h) dh = 1.

since by hypothesis V and W are irreducibels and therefore the L2-norm of their

characters equal one.We also used that χ[V �W ] = χπ

∗G

V χπ∗H

W , since characters are multiplicative, andthe obvious relation χπ

∗G

V (g, h) = χV (g) (and ditto for χπ∗H

W ) holds. ❏

Example ��. — The representations of tori. Let Tn = S

1×S1× . . . S

1 be an-dimensional torus, and let πi be the projection onto factor number i. Recall thatthe circle S

1 has the representations Lp given by z · c = zpc, which when p takes

every integral value, are all the irreducibles. It follows from proposition 9 that theirreducibles of T

n are of the form Lp1 � Lp2 � · · · � Lpnfor all choices p1, . . . , pn of

integers.On the level of representation rings, RCT

n = Z[l1, l−11 , .., ln, l

−1n

] where li denotesthe isomorphism class of the representation π

∗iL1. ❅

Versjon: Thursday, September 27, 2012 10:58:20 AM

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notes 6: representations — basics.€¦ · notes 6: representations — basics. version 1.00 —...

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