Representations of finite and Lie groups

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REPRESENTATIONS OF

Finitt and Lie Groups

This page intentionally left blank

Charles B Thomas University of Cambridge, UK

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Re. Ltd.

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British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

REPRESENTATIONS OF FINITE AND LIE GROUPS Copyright 0 2004 by Imperial College Press All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

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ISBN 1-86094-482-5 ISBN 1-86094-484-1 (pbk)

Printed in Singapore by World Scientific Printers (S)Pte Ltd

C.B. Thomas In memory of Ali Frohlich (1916-2001)


Preface

Good grief, not another book on representation theory! A cursory inspection of the small, if select, library at the Max-Planck-Institut in Bonn yields at least eight good introductory texts. These include the elegant book by J.P. Serre [J-P. Serre], against which all others should be judged. Beyond that the choice is perhaps a matter of taste - what particular slant does the author give to the subject, has she or he any special concerns? The approach chosen here is to present the elementary representation theory of finite groups in characteristic zero in a way which generalises immediately to compact topological groups. The only fresh ingredient needed is an invariant integral, which replaces taking the average by means of the sum over the group elements divided by its order. The parallel development is summarised at the end of Chapter 6 ; with finite groups as a special case of compact groups there is an inner product on the space of class functions under which the irreducible characters form a normal orthogonal set spanning a dense subspace. Two other topics receive special attention, exterior powers and the finite algebraic groups SL2(Pp). I have long believed that the A-structure of the representation ring R ( G ) is a much under-used tool. Some indication of this is given in the exercises devoted to the symmetric groups S,, but the applications are much wider, extending not only to the various families of simple groups of Lie type, but also to the 26 sporadic groups. As a topologist I have long been interested in SLz(lFp),and Chapter 8 is intended to illustrate the general principle that in characteristic zero the representation theory of a finite algebraic group has the flavour of the theory for the corresponding group defined over R or C. In contrast in the natural characteristic p the model is that of a maximal compact subgroup in the complexification.

vii

viii

Preface

The exercises are an important part of the text, and should be attempted, not just for their own sake, but also because in a few cases the results are used in a later chapter. The book concludes with an uneven collection of hints, worked solutions and additional references. The bibliography is short and contains no more than the rival books, which I have consulted, and references to theorems mentioned in the text but not proved. The starred sections (*) may be omitted at a first reading. The book has grown out of various sets of notes for a course of 16 or 24 lectures at the senior year level at Cambridge. My thanks are due to the generations of students who have attended, and interrupted, these lectures and to those who I have individually supervised. Their comments are a reminder of what a privilege it is to work in a great university. Errors inevitably remain, and are solely my responsibility. I wrote the final version during sabbatical leave from Cambridge at the University of California at Santa Cruz, Stanford University and the MaxPlanck-Institut in Bonn. I am grateful to all three institutions for their hospitality and support. I also thank Laurent Chaminade and Gabriella Frescura at Imperial College Press for their help, and most of all Michele Bailey for typing and producing the camera-ready text. Bonn, Michaelmas 2003

Contents

Preface

vii

1. Introduction

1

2. Basic Representation Theory - I

11

3. Basic Representation Theory - I1

25

.... ..... .........

30 31

4. Induced Representations and their Characters

35

5. Multilinear Algebra

47

Representations of GI x GZ * Real Representations . . .

Alternating and Symmetric Products . . . . . . . The Representation Ring R(G) and its A-structure * Representations of SZz(F,) in Characteristic p .

6. Representations of Compact Groups Induced Representations . . . . . Irreducible Representations of SU2

52 55 57 63

.., ..... ...... . .. ..... . .......... .

7. Lie Groups

67 69 75

Representations of the Lie algebra

8. SLz(IW)

.....

,

... ......

84

89 ix

*

.

Contents

Principal Series for SL2(IF,) . . . . . . . . . . . . . . . . . . . . Discrete Series for SL2(P,) . . . . . . . . . . . . . . . . . . . . . The Non-compact Lie Group SL2(R) . . . . . . . . . . . . . . Principal Series . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

91 92 95 97 97

Appendix A Integration over Topological Groups

101

Appendix B Rings with Minimal Condition

107

Appendix C Modular Representations

115

Solutions and Hints for the Exercises

121

Bibliography

143

Index

145

Chapter 1

Introduction

Our topic is the representation theory of finite and, more generally, of compact topological groups . The latter will be defined formally later; for the moment the reader can think of a topological group as a set carrying both a topology and a group structure, which are compatible in the sense that multiplication and inversion are continuous. Examples are SL2 (R)(noncompact) and the special unitary groups Sun (compact), both of which are important in theoretical physics. A representation of G is a homomorphism of G into Autc(V), the group of linear automorphisms of a finite dimensional vector space over the complex number @. By choosing a basis {el . . . en} of V such a representation determines a homomorphism

p :G

-

GLn(@).

If G carries a topology, we give GLn(C)its topology as an open subset of C n 2 ,and require the homomorphism p to be continuous.

Examples. (i) Let CF and let

{ a : a‘ = 1) be a cyclic group of order T generated by a , some primitive r t h root of unity. The homomorphism aq : C, .+ Ul C @* = C - (0) which maps a to CQ is a 1-dimensional representation of the group. Note that aq is injective (we say that aq is ‘faithful’) if and only the greatest common divisor (gcd = ( r ,4 ) ) of r and q equals 1. We will see later that every representation of a finite abelian group A is built up from ‘irreducible’ representations of this kind. (ii) Let G = Qs, the quaternion group with presentation =

C be

{ a , b : a4

=

1, a 2 = b2, b-lab = a-I}. 1

2

Representations of Finite and Lie Groups

Such a 'presentation' can be regarded as a contracted multiplication table in that it tells us that each element can be written as a product aiP, that there are eight such distinct products, and that they can be multiplied using the rule ba-' = ab repeatedly. [Exercise- Write out the multiplication table, and check that it corresponds to that of the basic quaternions {fl, f i , fj,fk} under the rule a H i , b j.] The map p

-

a-

(i"i)

b-

(yi')

extends to a homomorphism of Q8 into SU2. If W denotes the algebra of quaternions, note that W is 2-dimensional over C and that SU2 may be identified with the quaternions of unit length, using the representation just defined. The group Qs also has 1dimensional representations, obtained by composing the projection homomorphism

having Ker(?r) equal to the subgroup generated by a', with any one of the four 1-dimensional representations mapping a, b to f l . We label these as l,a,pand [email protected] the multiplication table we see that Q8 has five conjugacy classes of elements (1) (a2)(a, a - l ) ( b , b-l)(ab, a-lb).

Taking the trace of the representating matrices, and noting that the trace is constant on conjugacy classes, we obtain the following table

We ask the reader to check three things about this square array. First and most importantly, each row can be associated unambiguously with one of the representations described. Secondly, if we add the first four entries in each column to twice the fifth we obtain 8 (equals the order 1Q81) for column one and zero otherwise. Thirdly, even though the matrices describing p are

Chapter 1: Introduction

3

complex, the entries in the table are real. We shall see later that these are special cases of important general phenomena. Before leaving Q g let us present the generalised quaternion group of order 4t, which will be useful later, Q4t = { a , b : u~~=

l , a t = b2, b-lab = U-'}.

In terms of the quaternion algebra W a can be identified with eKiltand b with j . Given a homorphism of G into Aut@(V)we can think of G as acting on the vector space V via the map g.v = p ( g ) v for all v E V . More directly we can define such an action as a (continuous) map

G x V -V (g,v) - g . v

satisfying g l ( g 2 o ) = ( 9 1 9 2 ) ~and 1 . v = w for all g1,g2 E G and o E V . At least when G is finite, the G-action and C-action (scalar product) on V can be combined as a C[G]-action, where @[GIdenotes the so-called group algebra of the finite group G. Definition. The ring C[G] consists of formal linear combinations

It is a straightforward and tedious exercise to check that, with these definitions of addition and multiplication, @[GI is a ring, which is commutative if and only if G is an abelian group. Although we are primarily interested in the complex group algebra, it is important to note that the same definition holds with A equal to any commutative ring rather than A = C. We then obtain the group ring A[G], which has been much-studied both by topologists and number theorists. Examples. (i) Write out the multiplication table for the group ring

lF2[C2], where IF2 is the finite field with 2 elements, and Cz is a cyclic group of order 2.


4

(ii) If Cg is a cyclic group of order p ( p = prime), and [ is a primitive show that the map root of unity (for example e extends to a homomorphism of rings Z[Cp] -+ Z(,

4 = [cp(.), cp(4l

> 0 for U # 0.

Again appealing to elementary linear algebra we can extract a positive definite self-adjoint square root u of cp2 and write u = 9 u - l . The square root u can be expressed as a polynomial in cp2, so that u and cp commute. Therefore u2 = (cpu-')(cpu-') = cp2u-2 = 1. It follows that the C[G]module V splits as VO@ Vi,where VO= {v : uv = v} and V1 = {v : uv = -v}. The semi-linearity of u implies that VOand V1 are W-subgroups of V such that Vi = i h ( i = Since cp2 and u are both G-maps, so is u 0 and Vo is the required real form of V .

a).

With

x regarded as a complex valued function the character of V ~ WisG

x+x. Theorem 3.8 shows that there are three possibilities for an irreducible, finitely generated, @[GI-moduleV . (i) The character x is not real-valued in which case V ( Ris ~irreducible over RG, has twice the dimension of V , and character x y. (ii) The character x is real-valued, and an R-form Vo exists. In this case the complexified module is Vo @ &.

+

Chapter 3: Basic Representation Theory - II

33

(iii) The character x is real-valued, but there is no R-form VO.In this case V ~ WisGagain irreducible of twice the dimension of V ,and the character is 2x. For more on this subject, see Exercise 6 below.

Exercises. 1. Let p : G -+ GL,(C) be a representation with character x. Show that Ker('p) = {x E G : x(x) = n}. Show further that for any x E G Ix(x)l n, and equality holds if and only if p(z) = XI, for some

x E c.

and b < a >. The entry in the (2,2) position in a reflects the relation F l a b = a-1 in the given presentation. Note that the normality of < a > implies that the matrix representing a is diagonal. This is a useful general property, which will occur again for more complicated groups.

[-'

40


Replacing D2p by the quaternion group Qdp introduces the relations 1,aP = b2 in place of the first two above. The matrices are almost 0 -1 the same; a is mapped to q!l) with q = eTi/P and b to . The

a2P =

(:

( )

-1 in the (1,2) position reflects the fact that the extension of < a > by < b > is no longer split, i.e. that b4 = 1 rather than b2. We next prove a very useful result (Blichfeldt's Theorem) on the representations of groups of order p t ( p = prime).

Lemma 4.4 Let G be a non-abelian group of orderpt. Then there exists a normal abelian subgroup A, which contains the centre Z(G) properly.

Proof. First note that the centre Z(G) of a p-group is always larger than the identity. This follows by counting the conjugacy classes, the number of elements in each of equals the index of a centralising subgroup, and hence equals 1 or a power of p . Hence the identity cannot be the only element to belong to a singleton conjugacy class. The quotient group G/Z(G)still has order equal to a power of p , so that its centre Z ( G / Z ( G ) )is non-trivial. Pull back a cyclic subgroup of order p in this centre to G, obtaining an element a which commutes: with Z ( G ) ,and take A = (a,Z(G)). Theorem 4.5 Let V be an irreducible @[GI-module.If dim@V > 1 there exists a proper subgroup H of G and an irreducible @[HI-moduleW such that V = i,W.

Proof. First suppose that V is faithful, i.e. p : G -+ Aut(V) is injective. Under this assumption we claim that there exists HI C G such that HI contains the subgroup A from Lemma 4.4,and an irreducible @[H1]-space W with V = i,W. Consider V as an A-space. As such it is a sum of irreducible 1dimensional A-spaces. Let v generate one of these (with character $). If w generates another 1-dimensional A-space and has the same character $J,then with Q E A , a(X1v X2w) = $(Q)(X~ZI X2w). If W$ C V consists of all 1-dimensional A-spaces with character $J, then

+

+

Let W = W$ for some fixed character $. Assume that W exhausts V, i.e. that we need only one character. For v # 0 E V and g E G we would have that g-'v describes a 1-dimensionalA-space, which would again have

Chapter 4: Induced Representations and their Characters

41

to have character $. But for a E A a(g-'v)

= $(a)g-'v,

and

(gag-')v

= g$(a)g-'v

= $(a)v (because

$ ( a ) is a scalar).

This shows that a and gag-' have the same effect on v for all g E G. But A is strictly larger than the centre Z ( G ) of G, and for some pair of elements g , a gag-' # a. This would contradict V being faithful, and we conclude that more than one character $ is needed to describe the A-restriction of V. Indeed G permutes the W, transitively. We argue as follows: Let v E W,,a E A and g E G. Then 4 w ) = g(g-lag)v = g$(g-lag)v = $g(a)gv, where & ( a ) = $(g-lw). Thus g : W , H W,, and 9-l : W,, --t W,. The action is transitive, since we can sum over all g translates, obtaining V' = CW+,( for some &,) C V. Irreducibility shows that V' = V. For 9

the final step we fix W = W$,, (for some $1) and let H1 stabilise W , i.e. hW = W . be the subgroup containing A with the property that h E HI Note that H1 is properly contained in G since V # W+l. As in the original discussion of induced representations (compare the construction of V1 above), V is certainly obtained from W by translating this subspace by means of a family of H1-coset representatives. This sum is direct by choice of H1 as the isotopy subgroup of the action of G on the {W+}.The subspace W used in the induction must be irreducible, since if it were not, V = i,W would be reducible also. It remains to remove the restriction that V is faithful. Let GO= Ker(p : G -+ AutV) and = GIGO. The space V is certainly faithful for the quotient group GIGO. If V is not 1-dimensional, G is not abelian, and there exists H c G with V = i, W for some fi-space W . Let H be the inverse image of fi in G. As a subgroup H contains Go and W is an irreducible H-space. The definition of fi as an isotropy group implies that H has the same property, and copying the argument for the quotient group we see that V = a, W as a representation induced up from W using the cosets G I H . Recall that 0 elementary group theory shows that GIH 2 G/Go/H/Go.

*

h

h

h

Note as a complement to the argument that by repeating the construction we eventually arrive at a 1-dimensional representation of some proper subgroup of the p-group G, from which to start the induction process.

42


Non-abelian groups of order p 3 provide beautiful examples of both Blichfeldt’s Theorem and of the explicit construction of induced representation spaces. We recall that, up to isomorphism, there are two non-abelian groups of order p 3 , which for convenience we refer to as ‘metacyclic’and ‘elementary’. We again assume that p is odd; the case p = 2 (when G 2 0 8 or Qs) has been considered in Chapter 1. Denote the two groups concerned by P+ and P- with presentations

P- = { a ,b : up2= bp = 1,b-lub = u1+P}, and P+ = { a , b , c : ap = bp = cP = [c,u]= [c,b]= 1,[a,b]= c}. In both cases the centre equals the commutator subgroup of order p , giving an abelianised group isomorphic to the direct product of two cyclic groups of order p . Hence there are p 2 1-dimensional representations; the remaining irreducible representations are obtained by induction up from a normal subgroup of index p . As a check we note that P+ contains p 2 ( p - 1) conjugacy classes of elements. By direct calculation we see that in terms of matrices model p-dimensional representations are given by

+

and

Here C = e2.rri/Pand 7 = e2.rri/P2 are primitive roots, in the case of Pwe induce up from the subgroup generated by a, and in the case of P+ from (b,c). The remaining representations are obtained by taking powers of 17 and C respectively. The p-dimensional representations for P* are examples of so-called monomial representations. A group which satisfies the conclusion of Theo-

Chapter 4: Induced Representations and their Characters

43

rem 4.5 is called an M-group, a class which is certainly larger than groups of prime power order. Recall that a group is solvable if its derived series terminates:

where Gi = [Gi-l,Gi-l]. It is known that an M-group is solvable, see [L. Dornhoff, $151 but not every solvable group satisfies the M-condition. The best that can be done for an arbitrary group G, using the regular representation, is to show that each irreducible representation of G is contained in an induced representation for some subgroup H G. Our warning example is the so-called binary tetrahedral group T* , with presentation

The group T* maps onto the tetrahedral group T , with characters as described in a previous chapter, with a kernel which is central and generated by p 2 . The order is 24 and each irreducible representation of T also occurs for T*. Simple arithmetic shows that the remaining representations all have degree equal to 2; they are obtained as follows. Identify the special unitary group SU2 with quaternions z of unit length, and R3 as the subspace of EX4 consisting of purely imaginary quaternions y. The map y zyz-' for fixed z is linear and distance preserving, that is defines an element T ( Z ) E so3. As a homomorphism of groups T is two-to-one, with kernel consisting of {flz}. This subgroup of order two is central, and we have a commutative diagram.

T*

T

P

SU2

SO3

P

in which the lower horizontal arrow p denotes the irreducible; 3-dimensional representation of the tetrahedral group. The lifted homomorphism is seen to be irreducible, its restriction to the subgroup Q 8 is the standard representation first considered in the introduction, and restriction to (x) gives a representation with character w + w - l ( w = e2ai/3). (The two other irreducible representations have restricted characters equal to 1 w and 1 w - l respectively.) Since T* has no subgroup of index 2 (why?), the representation cannot be obtained by induction.

+

+

44


For more on the corresponding map n : SU2 compact groups.

t

SO3 see Chapter 6 on

Exercises.

1. Determine the irreducible representations of the dihedral group

and of the quaternion group

2. Let p be an odd prime and P& the two groups of order p3 , whose irreducible representations have been described in the main text. Prove the claim made that P+ and P- both contain p 2 ( p - 1) conjugacy classes, and write down the character table for each of the two groups. What do you notice? 3. Prove that if the order of the group G is a power pt of a prime p , where t 2 2, then the abelianisation G/[G,G] has order equal to at least p 2 . Prove further that every irreducible representation of a group of order p4 bas degree equal to 1 or p . 4. Let G = H x K be the direct product of two subgroups, and let p be a representation of G induced up from a representation 8 of H . Show that p is equivalent to 8 @ p r e g ,where ~ p r e g ,denotes ~ the regular representation of K . 5. Find all the characters of S5 induced from the irreducible characters of S 4 . Hence recover the character table of S5. Repeat, replacing S 4 by the subgroup ((12345),(2354)) of order 20 in S5. *6. (Maclcey 's criterion f o r the irreducibility of a n induced representation). Let W be the representation space for 8 : H + Aut(W) and V the induced representation space for p = i,0. If H, = H"2Hx-l we can distinguish between the representations 8" (conjugate by z) and ResH, (8) of the subgroup H,.. Let 8" : H , -+ Aut(W,). Prove that the space V is irreducible if and only if both the following conditions are satisfied:

+

(a) W is irreducible, and (b) for each x E G - H the representations 0" and ResH, (0)

are disjoint, i.e. have no summands in common.

Chapter

4 : Induced Representations and their Characters

45

[Hint: V is irreducible if and only if the inner product ( X V ,X V ) = 1. By F'robenius reciprocity this equals ( X W , i*i,xw), with the second inner product being taken over H. Now decomposes i*i, W as a direct sum of representation spaces Indg3:W, for a suitable family of elements 5. (This is an application of the so-called double coset formula.) The reader may find it easier to consider first the special case when H is normal in G.)] *7. Let G be the symmetric group S, and let X = { 1 , 2 , . . . ,n}. Write X , for the set of all r-element subsets of X , and let 7rr be the permutation character of the action of G on X , . If T s n / 2 show that G has r+ 1 orbits in its action on X , x X s , and deduce that ( 7 r r , n,) = r+l. It follows that the generalised character 7rr - 7 r r - 1 is irreducible for 1 T n / 2 . *8. Let G be the semidirect product of A and H with A normal in G and the subgroup H 2 G/A. This means that each element g E G can be written uniquely as a product g = ah, and that multiplication is twisted by the action of H on A. The dihedral groups D2, or D2t are examples which we have previously considered. We restrict attention to the special case when A is abelian. The group H acts by conjugation on the group Hom(A, C") of characters of A. If xi represents an orbit of this action, then we can form the product Gi = A . Hi where Hi stabilises xi. As a complex valued class function xi extends to Gi, and by composing with the projection map Gi .+ Hi for any irreducible representation 0 we obtain a representation 0' of Gi. Let pi,@= Ind& 8 0'). Prove

<
n. For the dimension count the family of subsets of { 1 , 2 . . .n).

54


By checking first on basis elements we see that A ( E ) is anticommutative in the sense that xy = ( - l ) d i m z ' d i m y Y X . We now turn to the symmetric analogue S ( E ) of A ( E ) . An rnmultilinear map f : E x . . - x E + F is said to be symmetric if

m

f ( ~ 1 , ... ,x,) = f ( x o ( l )., . . , x,,(,)) for all permutations (T E S,. In T m ( E ) let ,6 be the submodule generated by all elements of the form

for all xi E E and all CC

-

(T

E

S,.

Define S"(E) = T m ( E ) / b mand S ( E ) =

@ S m ( E ) ,the symmetric algebra of E. As in the case of alternating

m = n.._

maps the composition E x

.

xE

--t

T"(E)

--t

S m ( E ) is universal for

m

rn-multilinear symmetric maps. For a reason which is about to become obvious we write ~ 1 x 2 .. x., for the image of ( X I , . . . ,x,). Proposition 5.8 Let E be R-free with dimRE = n, and basis ( ~ 1 , ... , vn}. As elements of S 1 ( E )the vi are R-linearly independent, and S ( E ) is isomorphic to the polynomial algebra R [ v l , .. . , v,]. Proof. This is intuitively obvious since modulo b = @b,

m

we can iden-

tify elements which differ only in the order of the components xi, i.e. we can make the variables in Proposition 5.6 commute. More formally let tl . . . tn be independent variables over R and form the polynomial algebra R [ t l , .. . t n ] .Let P, be the submodule of homogeneous polynomials of degree m, and define a map E x . . . x E + P, as follows:

This map is multilinear and symmetric, hence factors through S"(E). The element w 1 . . . w, maps to tl . . . t , and similarly for each wi, . . . wi,. Linear independence of the monomials p ( ( ) ( t implies ) that this map is an isomorphism (obvious for rn = 1). By inspection the map is compatible with multiplication and grading.

Chapter 5: Multilinear Algebra

55

The Representation Ring R ( G ) and its A-structure The set of equivalence classes of @[GI-modules(@-vector spaces with Gaction) admits an addition (direct sum @) and a multiplication (tensor product 8 over C). Because of the results in Chapter 2, in particular Theorem 2.8 and its consequences, we can turn our structured set into a commutative ring R(G), called the (complex) representation ring of G. If it is necessary to emphasise the field of definition we write R(G) = RU(G). The elements of R(G) are formal differences [V]- [W] of equivalence classes of k

miV, with

modules; alternatively we can consider integral combinations i=l

mi E Z,rather than mi E Nu{O}. The real representation ring RO(G) is defined in the same way. These rings have a large amount of additional structure. If V is a @[GI-moduleand g E GI then since p ( g ) E Aute(V) we have an induced automorphism T"(g) : T m ( V )-+ T"(V) for each m. This makes both T"(V) and the direct sum T ( V )= @T"(V) into C[G]m

modules, with similar remarks applying to the symmetric and alternating powers. It is traditional to write Am(V) for the image of the alternating power A"(V), or rather of its equivalence class, in R(G). oc)

C

Notation: & ( V )=

Am(V)tm.

m=O

Strictly speaking one first defines At on the multiplicative monoid of equivalence classes of C[G]-modules, i.e. considers positive representations only, and then extends A t to all of R(G). In the end one obtains an exponential map At : R(G)

+

1

+ tR(G)[[t]]

of the abelian group R(G) into the multiplicative group of formal power series with constant term equal to 1. The fact that At is a homomorphism follows from

Lemma 5.9

@

( E ) @ r \ j ( F )E A ~ ( E @ F ) .

i+j=m

Proof.

0

Compare bases on the two sides. 00

Similarly we can define s t ( V )

=

C sm(V)tm1using

the symmetric

m=O

powers. For any C[G]-module V , we then have the non-trivial relation s , ( V ) L t ( V )= 1.


56

If dimCV = 1, &(V) = 1 relation follows since

+

(1 s l t

+ (Slt)2 +

+ s'(V)t,

* *

since Az(V)

=

0 for i > 1. The

.)(1- s l t ) = St(V)X-,(V)

=

1.

The general case may be reduced to a sum of 1-dimensional representations by means of the so-called 'splitting principle' for A-rings (see [M.F. Atiyah, D.O. Tall] or [D. Knutson]). The author believes that the full power of representation theory only becomes apparent when the alternating structure of R(G) is taken into account. A clue to this is provided by the following result for the symmetric group S, .

The exterior powers A k V of the standard (n - 1)dimensional representation V of S, are all irreducible, 0 6 k 6 n - 1.

Proposition 5.10

Proof. If x denotes the character of the permutation representation (add a trivial summand to V) then V will be irreducible provided that the inner product of x with itself equals 2. The same holds for AkV, since

we again write x for X A k @ n and will prove that (x,x) = 2. Let A equal the set {1,2,. . . ,n}. For a subset B of A with k elements and g E S, let 0 if g B

# B,

1 if g B = B and glB is even, and -1 if g B = B and glB is odd.

Then by looking a t a basis for the kth exterior power we see that x ( g ) =

C{g}B. B Therefore

Chapter 5: Multilinear Algebra

57

Here the sums are taken over subsets B and C of A containing k elements, except that in the last sum we neglect zero terms, and sum over those g with gB = B and gC = C. Such a permutation g splits into permutations of the four subsets B n C , B \ ( B n C ) ,C \ ( B n C) and A \ ( B U C). If C equals the numbers of elements in B n C rewrite the last sum above as

-cc c c c c 1

n!

B

c

aESe b E S k - e

CESk-e

(sgna)2(sgnb)(sgnc)(sgnd)2

dES,-nk+e

( s g n b) B

C

bESk-e

(sgnc)

CESk-e

The last two sums contribute zero unless k - C = 0 or 1,since otherwise the individual summands will cancel in pairs. If k = C, B and C coincide, and the expression reduces to $ C k ! ( n - k ) ! , B

5

which equals ( ; ) k ! ( n- k ) ! = 1. The terms with k - e = 1 similarly add up to 1, and required.

(x,x)= 2, as 0

If we refer back to the examples discussed at the end of Chapter 2 we see that S5 has seven irreducible representations, five of which are given by 1,V, A2V,E and E 18V, where E denotes the 1-dimensional determinant module. We know that V @V splits as A2V@S2V;the latter is a reducible module of dimension 10. This splits as U @ ( E @ U ) say. Assuming the relation st(V)X-t(V) = 1 we see that the character table can be built up from a knowledge of the character of V. In Exercise 3 below we invite the reader to consider SS in a similar way.

* Representations of SL2(P,)

in Characteristic p

As a lead-in to the representations of the compact group SU2 in the next chapter, consider the following application of symmetric powers to the IFpnrepresentation theory of the finite group SL2(IFP).Note that we must expect this to be totally different to anything so far considered, since we can no longer divide an expression by the prime p , and Maschke’s Theorem must be expected to fail. Let K be an algebraic closure of the finite field IF, and G = SLz(IF,), the group of 2 x 2 invertible matrices of determinant 1. We will construct p irreducible K[G]-modules Vm(O< m < p - 1) with dimK V, = m + 1. Let


58

Vm be the K-vector space of homogeneous polynomials of degree m in the independent variables x and y. If A = (aij) E G we put

This gives us a (right) representation module of the correct dimension, which we will show to be irreducible. Let (0) # W C V, and suppose that n

0 # f =zajxjy"-j

E Wl with a,

# 0 and n < m.

j=O

For t E IF, write

S ( t )=

(i I)

n

Then f S ( t ) =

C

uj(z

and T ( t )=

+ ty)jym-j

(: :).

n

=

C

f j ( x l y ) t j E W. Hence by

j=O

j=O

rearrangement we have a family of polynomials such that fo = f and f n = anym. W is closed under scalar multiplication by elements of IF,, so P- 1

C t - l ( fS(t)) E t=l In IF,

Ctt = t=l

I)-

1

t=l

w.

-1 if p - 1 divides i

0 otherwise] so that P-1

n

n

P-1

t=l

j=o

j=o

t=l

=i

-f i for 1

p-llj-2

< i

We illustrate these results by the important example of SU2. Recall the notation of the example at the end of Chapter 5 with the complex numbers @ replacing the algebraic closure Fp of IFp. As before VO is the trivial representation on C and Vl the standard representation on C2 (the operation being given by matrix multiplication). For m 3 2 let V, be the space of homogeneous polynomials of degree m in the variables x and y, so that dim@V, = m 1. Viewing polynomials as functions on C2 we obtain an SU2-action via (gP)a:= P(xy), where

+

Since g acts as a homogeneous linear transformation, the subspaces Vm C @[a:, y] are indeed SU2-invariant. As before the polynomials 6 Ic 6 m) form a basis for the vector space V., STEP 1. The representation space V, is irreducible. The argument used to prove Schur's lemma shows that it suffices to show that each linear SU2-equivariant map from V, to itself is a multiple of the identity. Let a be such a map and for each A E U1 set

and gxCYPk = Ck!gxPk = CYA2k-mPk = A2"-"CYPk. Then gxpk = X2",Pk Choose X so that all the powers Ask-,, 0 < Ic m, are distinct. For this is generated by P k . Since X P k belongs the ;\2k-m-eigenspace of gx in to this eigenspace CUPk = C k P k for some Ck E C.

v,

X (iv) [ [ XY, ] Z ]+ [[Y, Z ] X ]+ [ [ Z ,X ] Y ]= 0 (Jacobi identity).

+

78

Proof.


Apply the definitions.

0

These properties make I'(TM) into a real Lie algebra . Some other examples are (a) The space R3 with the 'cross product' as operation, (b) The algebra of real n x n matrices with [A,B]= AB - BA, and (c) Any real vector space with all brackets set equal to zero. A Lie algebra of type (c) is called commutative (or abelian). Lie subalgebras and Lie homomorphisms have the obvious definitions.

Definition. Let. cp : M" --t N" be smooth. The vector fields X on M and Y on N are said to be cp-related, if p* . X = Y ' cp. Lemma 7.4 If the pairs of vector fields {Xi, yi : i then [XI, XZ]and [YI , Yz] are also cp-related.

Proof.

=

1,2} are p-related,

For each f E Cm(W,R) we need to show that

[Xl,XZIz(f .cp)

=

[Yl,YZIrp(x)(f)r

which is done by unravelling the definitions.

0

Corollary 7.5 If X and Y are left-invariant vector fields for the Lie group G , then [X,Y] is also left-invariant.

Proof.

Apply 7.4 with cp = Lx.

0

This corollary shows that we may define the Lie algebra of G to be the subalgebra of I ' ( T M ) formed by all left-invariant vector fields. If we denote it by g we have already shown that g is isomorphic as a real vector space to TeG, the tangent space to the group a t the identity. In this way TeG acquires a Lie algebra structure, and clearly dim g = dim TeG = dim G. In showing that [X, Y]is indeed a vector field above we obtained an expression for the bracket operation on vector fields in terms of local coordinates, viz

Examples. 1. Let G = (Rn, +); then g is the abelian Lie algebra Rn with all brackets equal to zero (Exercise).

79

Chapter 7: Lie Groups

2. Let G = GL,(R); then TeG = M,(R) RnZ,which describes g as a vector space. To each X E g associate the (n x n ) matrix A = ( a z j ) of components of X ( e ) , so that X ( e ) =

Cat3(-&Ie),

and write A =

23

p ( X ) . Then by an explicit inspection of components one can show that p [ X ,Y ] = p ( X ) p ( Y )- p ( Y ) p ( X ) , giving the Lie algebra structure on B = eL(R). Here is an alternative definition of the Lie bracket, which can also be used to show that the vector field and matrix constructions agree on TeG. Let c(g) denote conjugation by g, x H gxg-l, and observe that the differential g* induces a linear isomorphism of TeG onto itself. This representation, written Ad : G + Aut (T,G) is called the adjoint representation, and in turn induces a homomorphism of Lie algebras

ad : TeG 4 Tl(Aut (TeG))= End (TeG). Given the use of conjugation in its construction it is not hard to see that the vector X is sent to the linear map Y H [ X ,Y ] ,or that

ad(X)Y = [X,Y]. (The truth of this formula can also be checked locally.) Let 'p : G + H be a homomorphism of Lie groups and denote their Lie algebras by g and b respectively. Define 'p* : g + b as follows: for each X E g, X ( e ) E T,G and the derivative assigns ' p * ( X ( e ) )E TeH to the vector X ( e ) . Here we may use either the definition of g in terms of tangent vectors or left-invariant vector fields.

Proposition 7.6 (i) If p : G -+ H is a homomorphism of Lie groups, then 'p. : g + l~ is a homomorphism of Lie algebras. (ii) the correspondence G H g, cp H cp* is a covariant functor from the category of Lie groups to the category of Lie algebras. Proof.

Granted (7.4) above this is a manipulative exercise.

0

Definition. The subset H of the Lie group G is a Lie Subgroup if (i) H is a subgroup and (ii) H is an immersed submanifold (with respect to some manifold structure).

80


Warning example: Define cp : R + T 2by cp(t)= (e2nit,e2niat), where cr is irrational. Then cp is an injective group homomorphism and an immersion. However cp(R) is a dense subset of T 2and is not embedded. If we require that H is embedded as a submanifold of G, then we have Proposition 7.7 Let H be a subgroup of G as a group, and a submanifold of G as a manifold. Then H is closed in G and is itself a Lie group with respect to the substructures.

Proof. The smoothness of the operations restricted to H is easy and is left as an exercise. In order to show that H is a closed subset of G it suffices to work near the identity e E G, and then use translation. But near e the pair (G, H ) is diffeomorphic to (Rm,Rkx 0), and any sequence of points {h, E H } may be considered as lying in Rk x 0, and hence converging to a limit z in this subspace. The point 5 thus belongs to H , which is closed. There is a harder converse to this proposition which says that if H is a closed subgroup of G, then H also has the structure of an embedded submanifold, and hence must be a Lie subgroup. As a special case the reader may like to try and prove Proposition 7.8 Let cp : G1 + G2 be a homomorphism of Lie groups, then cp has constant rank on GI. It follows that H = K e r cp is a closed, embedded submanifold of GI, and is a Lie subgroup of dimension equal to dim G - rank (cp*).

Hint: The constancy of the rank follows from a left translation argument, and in order to obtain a nice atlas for Ker (cp) we can appeal to a corollary of the inverse function theorem in order to find good local coordinates. Here are some examples of subgroups of GLn(R) and GLn(C), which are also submanifolds and hence closed in the general linear group concerned. The technique is to define the subgroup H in G by means of coordinate functions F i ( z l , . . . ,z,)

=0

(1 6 i

such that the rank of the Jacobian matrix

< m < dimG)

(z)

takes the maximum value

m at each point h E H . If this condition is satisfied, a variant of the inverse

function theorem allows us to choose new local coordinates for G with the vanishing of m of them defining H . The co-dimension of H in G equals m.

Chapter 7: Lie Groups

81

1. The groups SLn(R) and SL,(C) are defined as hypersurfaces (det X = 1) in open subsets of Rn2 and Cn2 respectively. 2. The orthogonal group 0, c GL,(R) is specified by the equations

which, allowing for symmetry, gives $n(n derivatives

+ 1)equations. Taking partial

+

Hence the minor (subdeterminant) of order $n(n 1) corresponding to the variables xst with s t does not vanish at 1,. The two component orthogonal group is a submanifold, and its dimension is - 1). The component which contains 1, is also a submanifold, SO,. A similar argument shows that U , is a submanifold containing SU, as a hypersurface.

, 0) for the group SVz construct,edin the previous chapter a.re irreducible and exhaustive. We do t.his by first. complexifying the algebra SUz. Let a be a. real Lie algebra with basis { e l , . . . en}, and Lie brackets [ei:ej] = C e i j k e k ( e i j k E R). k

As a vector space we hare a@ = C

@, a R

(compare Chapter 5). Extend

the bracket,s a.bove to ac by int.erpret.ingthe coefficients e i j k as complex numbers. The complex Lie algebra ac is called t.he complexification of a. Note that this definition is independent of the actual basis used. A similar argument shows t.hat we can complexify a representation space V for a (defined over t.he complex numbers!) to a representation space for ac. The original and ext,ended represent.ations share the same invariant subspaces. Each element of ac may be written iq(f,, v E a), and the extension of a representat,ion r from a t.o ay follows the rule

with 1 + w + w 2 0), and the element a is represented ). The reader knowing some topology may either by (i :!) or by ( o r3

group (b2 H

=

r2

recognise these representations, through which the group acts on the 3sphere, as defining Poincare's homology %sphere. The two orbit spaces are diffeomorphic. Here is an alternative construction of the representations 8j in the discrete series, which may appear to be more natural. As in our discussion of

94

E

10

c3

9)

+E

Y

0

I

4

E

4

+ -

4

v

I

c

4

Representations of Finite and Lie G T O U ~ S

0

+

-

0

*

*

i

Chapter 8: SLz(E%)

95

the representations of SL2(Pp)over the algebraic closure Fp in an earlier chapter, consider the ‘character’ of the natural representation V1 in the automorphisms of a 2-dimensional vector space. On the p’-elements (elements of order not divisible by p) we obtain the Fp-values

x

ae

z

I 1 2

bm

where y denotes an eigenvalue in Fp for the matrix b. Remember that b has been chosen to diagonalise over F P 2 but not over Fp. We now lift this Brauer character to a class function in characteristic zero by mapping Q and y to (p - 1) st and (p 1) st roots of unity respectively, i.e. to powers of p and 0. Note that choice is involved here, contributing to the range of values of i and j in the final character table. We must also make a consistent lift at the prime 2 in view of the relation a 9 = b q . We complete the definition of our class function by mapping the four p-classes in G in the same way as 12. At least when i and j are both odd we now have

+

X

12

z

c

d

xi -,gj

2

-2

2

2

bm

ae pie + p - i e

,im

+,-im

showing that the character 9, can be obtained by subtracting our lifted class function from xi. It remains to show that the class function is a virtual character. This uses a deep theorem of R. Brauer, which asserts that this is so, provided that we have a character on restricting the class function to ‘elementary’ subgroups. For an arbitrary group G these are direct products of cyclic subgroups with subgroups of coprime, prime power order. With G = SL2(Pp) such subgroups are built up from cyclic or quaternionic subgroups, and the answer is immediate. We now proceed to split Bp+l as ~1 r/2 in the same way as before. 2

+

The Non-compact Lie Group SLz(IR) In this section we aim to give some indication of how the representations of SL2(Pp),constructed using finite group theoretical methods, give a guide to the representations of SL2(R). So far as possible we will use the notation


96

already introduced.

D = diagonal subgroup

=

{(;a!!l)

:aER-{O)

U = unipotent subgroup = B = semidirect product U via the rule

>Q

D with D acting on the normal subgroup U

If = { Z E C : Im(z) > 0) denotes the upper half of the complex plane, there is an action of SL2(IR) on b given by

+

az b AZ = - w i t h A = cz d '

+

The matrices

f l 2

(::)

act trivially.

Let K be the isotropy subgroup of i, that is the subgroup of matrices = i. Then ai b = -c d i implies that we can take such that cos 9 - sin 9 a = d = cos9, b = -c = -sine, i.e. K =

+

+

The action of SL2(IR) is thus transitive, and there is a (2 - 1) map

We order the matrices in the product so that a H ca as y H 00 in g. We also allow K to act on the right of the space of cosets B\G, and give the space of complex valued functions L 2 ( K )the usual norm llfllk = J, If(lc)12dlc. Topologically we have shown that G decomposes as U x D x K , corresponding to the set decomposition of SL2(F,) into subsets of order p - 1,p and p 1. G also inherits a product measure from the hyperbolic measure d x d y / y 2 on B and dlc on K Z S1. As a topological space G is isomorphic to

+

0

a product of the open disc D2 and S' . This becomes even more transparent on replacing SL2(R) by the group SU1,l- see the Exercise 2 below.

Chapter 8: SLz(1W)

97

Principal Series We start with the representation of B given by G^,(ud)= a s , s E C. Let H ( s ) be the space of complex valued functions f on G, such that (i) f ( u d l c ) = a s f ' f ( k )and (ii) the restriction o f f to K belongs to L 2 ( K ) . The representation P s = G: is obtained by allowing g to act on H ( s ) on the right

P S ( g ) f ( s= ) f(29). This definition is a variant of that of an induced representation for a finite or compact group, but in this case H ( s ) is an infinite dimensional Hilbert space, and care is needed to show that P s is both bounded and unitary. This is so for s = iv(v # 0), a pure imaginary value, and H ( i v ) contains two irreducible unitary representation modules P+iiv and P-)i". Note the in the finite case. parallel with the splitting 51 + J 2 = GGw 2

For s = 0 there is a 3-fold splitting, only one component P+>Oappears in the principal series. Neglecting P->O leaves a third component, associated with what is called the mock discrete series . Neither this nor the complim e n t a r y series { s E (-1, I), s # 0, positive parity only} contributes to the 'regular representation' L 2 ( G ), with G acting by translation. In addition to Pfgiv we need a series corresponding to the 8-series in the finite case.

Discrete Series On we already have the volume element d p = d x d y / y 2 . Replace this by dpm = ym-2dxdy and let H ( m ) = L&,,(b,pm), m 2 2, be the space of holomorphic functions f : b 4 C which are square integrable with respect to the measure p m . If H ( m ) is given the usual Hermitian inner product via integration, this can be shown to be complete. If A-l

SLz(W) the representation D+i"

: G + Aut

D + ? " ( A ) f ( z )= f

(:I:)

=

(z i)

E

G=

H ( m ) is defined by

- (cz

+ d)-".

This can be shown to be unitary, and D-1" is defined in a similar way to act on the space of complex conjugates. At this point recall that if

I


98

p = 3(mod4) the character O(+) splits as 771 +qz with 'discrete series' is explained by the following result:

The representation D+>rnis irreducible.

Theorem 8.1

(z) n

772 = 5jl.

+

The name

If gn(z)

=

(Z i)-n, gn E H . If Hm+an is the subspace spanned by $ J ~ , Hrn+an is an eigenspace f o r the compact subgroup K , and

It remains to give some explanation of why only the principal and discrete series appear in the regular representation of SL2(W). We do this by giving a measure j on the 'space' SLz(IW)" of irreducible representations. For the positive and negative parts of the discrete series dj = (this point weighting corresponds to a distributional character), and for the principal series d j ( v + ) = t a n h ( y ) d v , dj(v-) = c o t h ( 7 ) d v . The remaining irreducible representations belong to set of measure 0. We illustrate this by the diagram below, and adjoin a corresponding diagram for SLz(F,). In this quick trip through the representation theory of SLz(IW) we have relied mainly on S. Lang's book [S. Lang (1993)l.

&;

&$

measure zero

SL2(W Fig. 2

Exercises 1. Determine the character table of SLZ(F)where F is a finite field containing q = 2t elements. (First show that S L 2 ( F ) contains q 1 conjugacy

+

Appendix A : Integration ower Topological Groups

99

characters of type classes, and determine their orders. Then find 0,. Again xo splits as 1~ $.) characters of type xi and

2. If S U l J = {

(: :)

+

:

a,b E C,laI2 - JbI2 = l}, show that SL2(R)

and SU1,l are isomorphic subgroups of GL2(C) by conjugating with the matrix A =

z

-i).

-,k+%)that SU1,l (2-i)

H

(t

0

Show further, using the Mobius transformation is the group of isometries of the Poincar6 model 0

D2 of the hyperbolic plane. Draw a picture of SL2(R) D2 x S1,and distinguish between elliptic (Itrace1 < 2), parabolic (Itrace1 = 2) and hyperbolic (Itrace1 > 2) subsets. 3. (For those with some knowledge of differential geometry) If exp is the exponential map introduced in the previous chapter, show that exp is surjective for GL2(C) but not for SLz(R). 4. Show that SL2(R) has no non-trivial finite-dimensional unitary representation.


Appendix A

Integration over Topological Groups

In discussing representations of compact groups we introduced the notion of the Haar integral, proving its existence for our motivating example of SU2. Here we give one of many proofs of its existence in general, using an argument which may remind the reader of the definition of the Riemann integral as the limit of a sequence of approximating finite sums. Another argument, which is often presented in books on Lie groups (see for example [Th. Brocker, T. tomDieck]) starts by integrating a function with support inside some small open subset modelled on R". The change of variable formula from advanced calculus ensures that this is well-defined. A globally supported function can be split by means of a 'partition of unity' into a sum of functions fi, each of which has support inside some Euclidean neighbourhood, and the integral of f is defined as the sum of the integrals of the fi. Group invariance follows if we first define the basic volume element in a neighbourhood of the identity e and then translate this to a neighbourhood of an arbitrary element g. We prefer not to do this, partly on aesthetic grounds, but mainly because generalising the basic results for representations of finite to compact groups does not require the latter to be locally Euclidean. Indeed the most general result known to the author only requires the group G to be locally compact and Hausdorff. Given a compact topological group G we shall show that there exists a unique rule for integrating a continuous function f. The integral should be a linear functional on the space of functions V , and should be non-negative if f is non-negative. In addition it should be left invariant, i.e.

f(h-lg)dg =

/

f(g)dg for all h E G,

G 101

(A.1)


102

and be normalised by

s,

(A.2)

1 . d g = 1.

If in addition it satisfies

we shall call our integral right invariant. Our first proposition is a version of the mean ergodic theorem in functional analysis, see [S. Sternberg].

Let T be a linear transformation of a nonned linear

Proposition A . l space V such that

(i) llTnull < cJJwIIfor some constant c, and (ii) there exists w E V for which the sequence Snw=

1

-

n-l

)

~ T ' W

n (+O

.

possesses a convergent subsequence with limit i5. Then T Z the sequence Snw converges t o a.

Proof.

The subspace ( I d - T ) V = { z E V : Snz

+

=

a and

0). Firstly, if

z = (Id - T)w,then

which tends to zero with increasing n. But ( i ) implies that the subset { z 6 V : Snz 4 0) is closed, so that every element z of the closure of (Id - T ) V satisfies S,z -+ 0. Therefore (Id - T)V { z : Snz 0). Conversely let Snz + 0, so that given E > 0 we can find no such that, for n 2 no ---f

11.2 -

( z - Snz)I\ < E .

But

1

+ (Id - T 2 ) z+ . . + (Id - T"-')z) - T ) { z+ (Id + T ) + . . . + (Id + T + . . . T " - 2 ) ~ ) ,

z - S,Z = -{(Id n =

1

-(Id n

-T)z

which is an element belonging to (Id - T ) V .

*

Appendix A: Integration over Topological Groups

If Sn, w

-+

for some subsequence {n1,722, . . . } of 1

TS,,w-Sn,w=(--)(Tn3w-w)

103

N,then

-+O, s o t h a t m = w .

Now

T"w = T"0 + T"(w - 0 )= Ur + T"(w s,w = w Sn(W - w).

+

By assumption (ii) w

-

S,, w

+w -0

- 0 ) ,so that

and we have proved that

w - Sn, w E ( I d - T ) K Therefore w - w E ( I d - T ) V , so that Sn(w under S, , the proposition is proved.

-

W ) 4 0. Since ;iiT is fixed

We wish to apply this to the space V of continuous complex valued functions on the compact group G, carrying the sup norm, i.e. l l f l l = sup If(g)1. As a compact group G is also totally bounded, i.e. there exists a basic neighbourhood system a t el {Ua : i E I } , and for each i a finite number of elements 91,.. . ,g3(a)such that G C U skuz.Take all the gk'S which arise as i varies and arrange them as some sequence {gk} which will be dense in G. Define an operator T on V by

) If. ( f I

-

f(y)l

< E for all z - l y

E U , where

U is some neighbourhood of el

then

3. For the moment note that our calculations suggest that there should be q l-dimensional representations and a family of q-dimensional irreducible +.. .), ~ ( b=) representations, with typical character x ( a ) = (C+Yl, = [z, [Y,

4

= a d k , Yl(Z).

[adz,ady] = ad[z,y].

Therefore (ii) Let e =

(: i) ,f (: :) ,h (i =

=

1 :)

. Then iil2

= ( e ,f , h, ) and

[ h , e ]= 2e, [ h , f ]= -2f and [ e , f ] = h. So

ade(e) = 0 , a d f ( e ) = -h,

a d e ( f ) = h, adf(f) = 0, a d h ( f )= - 2 f ,

adh(e) = 2e,

ade(h) = -2e, adf(h) = 2 f , adh(h) = 0.

Hence the matrices representing ade,adf and adh with respect to the basis { e , f , h} are respectively

0 0 -2

0 0-0

2 0 0

Call these matrices E , F and H . Now 503 = ( A ,B , C , ) ,where

0 -1 0

0 0 -1

00 0

and [A,B] = C , [ A ,C] = -B and [B,C] = A. It is sufficient to find a matrix P E GL3(@)with the mapping (pp : { E ,F, H } 4 A&(@), X P - l X P having { A ,B , C} as its image. 3. Let {vi} and {wj}be bases for V and W respectively. ---f


138

+

+

(a) We show that y(s y) = y(z) y(y) by applying the left-hand side to the generic basis element vi 8 wj. A similar argument applies to

Y(XX). (b) [y(z),y(y)l = (ZY - YZ) 8 I 8 ( 2 ’ ~ ’- ~ ’ 2 ’= ) y([a,y]) by formal manipulation. (c) Write cp : g + End(V 8 W) for the map z H p(z) 8 p ’ ( y ) , and take p = p1 and p’ = p 2 to be the usual representations of degrees 2,3 of 5I2. Then with the same notation as in Exercise 2,

By calculating the effect of p(H), p(E) o p(F) and p(F) 0 cp(E) on the basis element

(i)

8

(!)

we see that

Hence cp is not a homomorphism of Lie algebras.

4. Given the solution of Chapter 6, Exercise 2, and the pairing of representations of SU2 and 5I2 in Corollary 7.10, we have only to note that the algebra and group representations are compatible with the exponential map. Thus

n times

5. Parts (i) and (ii) are routine calculation. For part (iii) we note that, by Schur’s lemma, the Casimir operator is constant on V,, and this constant may be computed by applying C to v. Here v is a ‘highest weight vector’, that is an eigenvector for H with maximal eigenvaluelweight. * (iv) We use the Casimir operator C to show that V is a direct sum of irreducibles. Assume that C has only one eigenvalue on V .


139

Let V * be a subspace of V that is a direct sum of irreducibles of maximal possible dimension. If V * # V find an irreducible subspace of V/V'. Because of the eigenvalue assumption for C, and the irreducible summands of V * are all isomorphic to some Vd. Here we use the same notation as in Chapters 6 and 7, together with elementary facts about semisimple modules from Chapter 1. Suppose that = U/V*. Then E$+'V = 0, so that E$+'U c V * . On the other hand E$+'V* = 0, hence E+ is nilpotent on U , say E?+' = 0, E$ # 0 on U . Thus k 2 d. Since [E;", E-] = (Ic -t1)(H- Ic)E$,Ic must be an eigenvalue of H , so that k < d. Therefore k = d. Now choose u E U such that ;ii is a highest weight vector in Let w = Edu and u' = E f w . Then 21' is a non-zero multiple of ii. On the other hand E+u' = 0, and we have just seen that u' = E f w is an eigenvector of H . It follows that u' generates a copy of V d not contained in V " ,a contradiction. See also [W. Fulton, J. Harris, page 4811. 6. p : SL,(@) --+ Aut(sI,(@)). (i) For all X E SL,(C), Y E sI,(C), truce ( X Y X - l ) = trace(Y). Moreover p ( X ) is linear, so p ( X ) E Aut(sI,(@)). Let X,2 E SL,(@), then under p ( X 2 ) Y is mapped to X Z Y Z - l X - l , so that p ( X 2 ) = p ( X ) o p ( Z ) ,i.e. p is a group homomorphism. (ii) As a vector space sK,(C) has dimension n2 - 1, since the condition This equals the dimension of the trace = 0 gives a hypersurface in PZ. domain group SL,(@), and if the representation were to split we would have a contradiction. (Any such splitting would give a representation of lower dimension than p , hence one with non-trivial kernel.) For more on the adjoint representation see [Th. Brocker, T. tomDieck] or [W. F'ulton, J. Harris].

v

v

u.

Chapter 8

1. With the same notation as for odd primes p write

where CY is a generator of the cyclic group F - (0). As before by passing to a field with q2 elements we can find a group element b of order q f 1.


140

The conjugacy classes and their orders are given by

Check: and o again denote primitive (q )– 1)st (q+1) + 1)st roots of unity the character table may be summarised as

I lcl $

12 1

xa

Q q+ 1

9,

q-1

I

I

ae 1 1 1 0 1 1 pie +p-ie -1 0

b" 1

C

-1 0 +j"

+ (T-j" )

As the entries betray, the construction proceeds as for p = odd. In splits as 1 + and for the family particular the induced character we use the induced charater and the relation

2. With

by direct calcula-

tion. Properties of Mobius transformations show that the spaces map upper half-planc goes to the boundary point as claimed; note that For the picture we have

SL2(R)

o2x S1, with copies of D2 normal to the page.


141

3. From the diagram above one has that the elliptic elements equal the union of all subgroups of SLz(R) which are isomorphic to the circle group S1. Those elements at the ‘top’ of the diagram (trace 2 2) form the union of all subgroups isomorphic to R. The remaining elements (trace 6 -2) do not belong to any 1-parameter subgroup, with the exception of -12. To see that the exponential map is surjective for GLz(cC) (indeed for GL,(C)) use the Jordan normal form. The diagonalisability of a unitary matrix can be used to show that ‘exp’ is surjective for U,. Like other properties this generalises to other compact Lie groups; any geodesic passing through the identity is infinitely extendable, and any two points can be joined by a geodesic. 4. The claim follows from the following three steps:

(i) With m any natural number and A ( t ) =

:(

A(t)

(%’

0)

= A(m2t)= A@)”’

(ii) Let cp be a finite-dimensional unitary representation, cp : SL2(R) -+ U,. By step (i) the eigenvalues of cpA(t)are a promutation of their m2-powers for any value of m and hence are roots of unity. They must therefore all equal 1. (iii) The representation cp is trivial on the normal subgroup generated by the matrices A@),which coincides with SLz(R) itself.


Bibliography

M.F. Atiyah: Power operations in K-theory, Quarterly J. Math. 17 (1966) pp. 165-193. M.F. Atiyah, D.O. Tall: Group representations,X-rings and the J-homomorphism, Topology 8 (1969), pp. 253-297. N. Blackburn, B. Huppert: Finite groups I1 (Grundlehren, no. 242) SpringerVerlag (1982). Th. Brocker, T. tomDieck: Representations of compact Lie groups, SpringerVerlag (1985) M. Burrow: Representation theory of finite groups’, Academic Press-NY (1965). R. Carter, G.B. Segal, I. Macdonald: Lectures on Lie groups and Lie algebras (LMS Student Text, no. 32), Cambridge University Press (1995). L. Dornhoff Group representation theory, Part A, Marcel Dekker-NY (1971). W. Fulton, J. Harris: Representation theory, a first course, Springer-Verlag (1991). G. James, M. Liebeck: Representations and characters of groups, Cambridge University Press (1993). J.L. Kelley: General topology, Springer-Verlag (Reprint of 1955 edition). D. Knutson: A-Rings and the representation theory of the symmetric group, Springer LN no. 308 (1973). S. Lang: Algebra, Springer-Verlag (3rd edition). S. Lang: SLz(W), Springer-Verlag (1993). J-P. Serre: Linear representations of finite groups, Springer-Verlag (1977). S. Sternberg: Group theory and physics, Cambridge University Press (1994). E.B. Vinberg: Linear representations of groups, Birkhauser-Base1 (1989).

143


Index

group ring, 3 Groups C,., cyclic, 1 Lhrn,dihedral, 10, 39, 44 I , icosahedral ( A 5 ) ,21 0, octahedral ( S 4 ) , 20 PSLz(lF7),23, 126 P*, order equals p3, 42, 44 Q 4 t , quaternion, 44 SLz(Iw), special linear, 1, 91 SLz(F,), characteristic 0 representations, 89, 92, 95, 123 SLz(F,), characteristic p representations, 57 SUz, special unitary, 2, 43, 64, 84, 86 5’5, symmetric, 44, 124 SS,symmetric, 127 T , tetrahedral (A4), 20 T ’ , binary tetrahedral (SL2(lF3)), 43 Un,unitary, 63, 81, 104, 105

pregular (p’-element), 116 A-structure, 55 1-parameter subgroups, 81 abelian group, 15, 26 adjoint representation, 79, 86 algebraic integer, 27 alternating product, 52 alternating square, 14 Brauer character, 116 theorem, 95 Burnside’s theorem, 9 Casimir operator, 87 character, 13 table, 19-21, 23, 93, 124 class function, 13, 18 Clebsch-Gordan formula, 71 complementary series, 97 complexification, 84 differential manifold, 75 discrete series, 92, 97 double centraliser condition, 8 double coset formula, 45, 131

Haar integral et seq., 101 indecomposable, 5 induced representation SLZ(W),97 finite, 35 compact, 67 irreducible, 1

exponential map, 82 F’robenius reciprocity, 38 145

146

isotypic, 26, 67 Jacobson radical, 119 Lie algebra, 78 5 [ 2 , (representations), 84 Lie group, 75 Mackey’s criterion, 38 Maschke’s theorem, 5 metacyclic group, 42 Method of little groups (Mackey-Wigner), 132 minimal condition, 107, 109 mock discrete series, 97 modular representation, 115 orthogonality relations, 15 principal series, 91, 97 product representation (GI x G z ) , 30 radical, 109 radical (nilradical), 119 real representation, 31, 33 F’robenius-Schur Theorem, 34

Index

regular representation, 17 representation, 1 representation ring R(G),55 representations of SU2, 57 Schur’s lemma, 5 semisimple module/ring, 4, 112 simple ring, 6, 108, 114 symmetric group S, exterior powers, 56 permutation representation, 20 symmetric product, 52 symmetric square, 14 tangent bundle/vector, 75 tensor product (advanced), 47 (utility), 12 topological group, 1, 63 Wedderburn’s theorem, 9 Young diagram, 130

Representations of finite and Lie groups

Representations of finite and Lie groups

Representations Of Finite And Lie Groups