Other Titles in This Series

10 Barry Simon, Representations of finite and compact groups, 1996

9 Dino Lorenzini, An invitation to arithmetic geometry, 1996

8 Winfried Just and Mart in Weese , Discovering modern set theory. I: The basics,

1996

7 Gerald J. Janusz, Algebraic number fields, second edition, 1996

6 Jens Carsten Jantzen, Lectures on quantum groups, 1996

5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995

4 Russel l A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994

3 Wil l iam W. A d a m s and Phi l ippe Loustaunau, An introduction to Grobner bases,

1994

2 Jack Graver, Brigi t te Servatius, and Herman Servatius, Combinatorial rigidity,

1993

1 Ethan Akin, The general topology of dynamical systems, 1993

http://dx.doi.org/10.1090/gsm/010

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Graduate Studies in Mathematics

Volume 10

Representations of Finite and Compact Groups

Barry Simon

V

American Mathematical Society

Editor ia l B o a r d

James E. Humphreys Lance Small

2000 Mathematics Subject Classification. Primary 20C05, 22E46; Secondary 20C30, 22E15.

ABSTRACT. This book is a comprehensive pedagogical presentation of the theory of representation of finite and compact Lie groups. We discuss both the general theory and representation of specific groups. Types of groups whose representation theory is discussed include finite groups of rotations, permutation groups, and the classical compact Lie groups. Along the way, the structure theory of the compact semisimple Lie groups is exposed. The approach tends to be that of an analyst.

Library of Congress Cataloging-in-Publicat ion Data

Simon, Barry, 1946— Representations of finite and compact groups / Barry Simon.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339; v. 10) Includes bibliographical references and index. ISBN 0-8218-0453-7 (alk. paper) 1. Representations of groups. 2. Finite groups. 3. Compact groups. I. Title. II. Series.

QA176.S56 1996 512'.2—dc20 95-42958

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams.org.

© Copyright 1996 by BARRY SIMON.

Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http://www.ams.org/

10 9 8 7 6 5 4 3 13 12 11 10 09 08

To the memory of my father, Hy Simon

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CONTENTS

Introduction xi

CHAPTER I. Groups and Counting Principles 1 1 Groups 1 2 G-spaces 2 3 Direct and semidirect products 5 4 Finite groups of rotations 11 5 The Platonic groups 13 6 The Sylow theorems 16 7 Counting and group structure 18

CHAPTER II. Fundamentals of Group Representations 21 1 Definition and unitarity 21 2 Irreducibility and complete reduction 23 3 The group algebra and the regular representations 25 4 Schur's lemma 27 5 Tensor products 29 6 Complex conjugate representations; Quaternionic representations 30 7 One-dimensional representations 34

CHAPTER III. Abstract Theory of Representations of Finite Groups 35 1 Orthogonality relations and the first fundamental relation 36 2 Characters, class functions, and conjugacy classes 39 3 One-dimensional representations 42 4 The dimension theorem 43 5 The theorem of Frobenius and Schur 47

Appendix to III.5—Representations on real and quaternionic vector spaces 50

6 Representations and group structure 55 7 Projections in the group algebra 56 8 Fourier analysis 57 9 Direct products 59

10 Restrictions 59 11 Subgroups of index 2 60 12 Examples 62

viii CONTENTS

CHAPTER IV. Representations of Concrete Finite Groups. I: Abelian and Clifford Groups 65

1 The structure of finite abelian groups 65 2 Representations of abelian groups 67 3 The Clifford group 68

CHAPTER V. Representations of Concrete Finite Groups. II: Semidirect Products and Induced Representations 77

1 Frobenius theory of semidirect products 77 2 Examples of the semidirect product theory 81 3 Induced representations 83 4 The Frobenius character formula 85 5 The Frobenius reciprocity theorem 89 6 Mackey irreducibility criterion 91 7 Semidirect products, revisited 93

CHAPTER VI. Representations of Concrete Finite Groups. Ill: The Symmetric Groups 95

1 Permutations and classes 95 2 Young frames and Young tableaux 96 3 Projections in A(Sn): Classification of representations 101 4 Branching relations 108 5 The Frobenius character formula 109 6 Consequences of the character formula 117

CHAPTER VII. Compact Groups 121 1 C°°-manifolds: A review 121 2 Lie groups and Lie algebras 128 3 Haar measure on Lie groups 133 4 Matrix groups 135 5 The classical groups 137 6 Homotopy and covering groups 146 7 Spin groups 152 8 The structure of compact groups 155 9 Representations of compact groups: Abstract theory 155

10 The Peter-Weyl theorem 158

CHAPTER VIII. The Structure of Compact Semisimple Groups 165 1 Maximal tori 165 2 The Killing form 170 3 Representations of tori 173 4 Representations of SU(2) and sl(2, C) 174 5 Roots and root spaces 177 6 Fundamental systems and their classification 183 7 Regular and singular elements 189

CONTENTS ix

8 The Weyl group 192 9 The classical groups 196

CHAPTER IX. The Representations of Compact Semisimple Groups 205 1 Geometry of the Cartan-Stiefel diagram 206 2 The geometry of integral forms 210 3 The Weyl integration formula 213 4 Maximal weights 215 5 The classification theorem and the Weyl character formula 217 6 Consequences of the Weyl character formula 219 7 Representation theory: The algebraic approach 225 8 Representations of the classical groups 227 9 Determinant formulas for the classical characters 237

10 Real and quaternionic representations of the classical groups 242 11 Tensors, permutations, and the Frobenius character formula 246

Appendices A Multilinear algebra 253 B The analysis of self-adjoint Hilbert-Schmidt operators 257

Bibliography 261

Index 263

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INTRODUCTION

Oy! YABOGR—yet another book on group representations. The theory is so beautiful and so central to mathematics that it tends to draw authors like honey draws flies. Given the existence of several excellent monographs (among which I'd include Adams [1], Fulton-Harris [5], Samelson [17], and Serre [19]), why do I feel this book a worthy addition to the textbook literature on the subject?

I think two facets distinguish my approach. First, this book is relatively ele­mentary; and second, while the bulk of books on the subject (and, in particular, the four quoted above) are written from the point of view of an algebraist or a geometer, this book is written with an analytical flavor.

As for the elementary nature of the material, much of it is self-contained. A prior exposure to the notions of quotient group and the isomorphism theorems is assumed; but, for example, I develop the necessary theory of algebraic integers in proving that the dimension of an irreducible representation divides the vector of the group. The material on finite groups (Chapters I-VI) should be suitable for an upper-level undergraduate course, either as a separate course or as a supplement to an advanced algebra course.

The material on compact groups is a little more sophisticated but I have discussed the needed calculus on manifolds and have even included an appendix on the basic theory of self-adjoint Hilbert-Schmidt operators, which is needed to prove the Peter-Weyl theory.

However, there are a few places we need some facts from algebraic topology that we used without proof: for example, the exact sequence of a fibration.

As for the analyst's point of view, much of the most profound work on group representations has been done by analysts: Weyl, Gel'fand, and Mackey come to mind. Indeed, this monograph bears a strong influence of George Mackey from whom I first learned much of the material thirty years ago.

The analyst's approach can be seen in several places: for example, the last three chapters discuss the structure and representations of the compact groups, not the representations of the semisimple Lie algebras. The two are closely related but the former is more elementary and decidedly less algebraic. In this regard, the discussion is closest to that of Adams [1].

A critical role is played by the fact that for compact groups, it is easy to show that any (finite-dimensional, continuous) representation supports an invariant positive definite inner product. This immediately implies that on the Lie algebra, the adjoint representation is by matrices which are skew-adjoint in a suitable inner product, so the operators are semisimple and the Killing form is (strictly) negative definite. This replaces pages of algebraic minutiae.

XI

X l l INTRODUCTION

A good example of this philosophy is the proof in Chapter VIII that all maximal tori in a compact Lie group are conjugate and the union of all the maximal tori is the entire group. The standard proofs either go through the conjugacy of the Cartan subalgebras and considerable additional argument, or else uses Weil's approach of using the Lefschitz fixed point theorem, a sledgehammer for what is a rather simple result. Instead, I use a simple argument inductive in the dimension of G. The first 90% of the proof is that used by Varadarajan [20], but at a critical point, he appeals to the structure theory of the semisimple Lie algebras, which requires tens of pages of careful algebraic argument. Instead, I use the existence of an invariant inner product for the adjoint representation.

This is one of dozens of places where the proofs are ones I found while polishing the book. Nevertheless, I am not so naive as to think that there are any proofs here that don't appear somewhere in the literature, which is vast. But I do claim a coherent, elementary approach.

Individual chapters begin with brief summaries of what they contain. Chapter I sets the stage, focusing on counting principles as a leitmotif. The high point is the Klein-Weyl determination of the finite subgroups of three-dimensional rotations.

Chapters II-VI discuss the representations of finite groups. Chapters II—III develop the general theory and Chapters IV-VI, the representations of specific families of groups: Abelian and Clifford groups in Chapter IV, semidirect products in Chapter V, and permutation groups in Chapter VI.

The final three chapters discuss the representations of compact groups, primarily compact Lie groups. Chapter VII discusses the general theory of Lie groups and the analogs of the results of Chapter III. Chapter VIII discusses the structure theory of compact Lie groups: maximal tori, roots, and the Weyl group. It is preparation for the final chapter which presents Weyl's theory of the representations of the classical groups. The final section draws together the two halves of the book in a fascinating way by providing a proof of the Probenius character formula for the permutation group.

By focusing on finite and compact groups, we can present the basics completely. Any attempt to go beyond this would yield a multiple-volume work (as it has in other cases!). Nevertheless, when I've given this as a one-year graduate course, I have spent five weeks discussing related topics from the representation theory of noncompact groups.

This book is based on a course I first gave at Princeton in the mid-70's. Over the ensuing twenty years, I have given the course roughly a half-dozen times at Princeton or Caltech, and each time additional polish was added. I am grateful to all the students in those courses for the feedback and insight they provided.

Parts of the actual manuscript were written during stays at the ETH-Zurich, Hebrew University, and the Technion. I appreciate their hospitality and, in par­ticular, the courtesy shown to me by C M . Graf, W. Hunziker, M. Ben-Artzi, and J. Avron. The preparation of the manuscript, which involved taming both TeX and my handwriting, was well handled by C. Galvez, to whom I'm grateful. I benefited from a careful reading and comments from S. Miller.

I hope you will enjoy this book. I can't think of any other course of mathematics with so much innate beauty so close to the surface.

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BIBLIOGRAPHY

[I] J.F. Adams, Lectures on Lie Groups, W.A. Benjamin, New York, 1969. [2] K. Chandrasekharan, Arithmetical Functions, Springer-Verlag, Berlin, 1970. [3] B. Eckmann, Gruppentheoretischer Beweis des Satzes von Hurwitz-Radon uber die Kompo-

sition quadratischen Formen, Comment. Math. Helv. 15 (1942/43), 358-366. [4] H.D. Fegan, Introduction to Compact Lie Groups, Series in Pure Mathematics, vol. 13,

World Scientific, Singapore, 1991. [5] W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Math­

ematics, vol. 129, Springer-Verlag, New York, 1991. [6] M.J. Greenberg and J.R. Harper, Algebraic Topology: A First Course, Mathematics Lecture

Note Series, vol. 58, Addison-Wesley, Redwood City, Calif., 1981. [7] S.-T. Hu, Homotopy Theory, Academic Press, New York, 1959. [8] I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [9] J.P. Jans, Rings and Homology, Holt, Rinehart and Winston, New York, 1964. [10] T.Y. Lam and T. Smith, On the Clifford-Littlewood-Eckmann groups: A new look at peri­

odicity mod 8, Rocky Mountain J. Math. 19 (1989), 749-786. [II] D.E. Littlewood, Note on the anticommuting matrices of Eddington, J. London Math. Soc.

9 (1934), 41-50. [12] L.H. Loomis and S. Sternberg, Advanced Calculus, Addison-Wesley, Reading, Mass., 1968. [13] J.F. Price, Lie Groups and Compact Groups, London Mathematical Society Lecture Note

Series, vol. 25, Cambridge University Press, Cambridge, 1977. [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis,

Academic Press, New York, 1972. [15] , Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness,

Academic Press, New York, 1975. [16] W. Rudin, Fourier Analysis on Groups, Tracts in Mathematics, no. 12, Interscience Pub­

lishers, New York, 1962. [17] H. Samelson, Notes on Lie Algebras, Van Nostrand Reinhold Mathematical Studies, no. 23,

Van Nostrand Reinhold, New York, 1969. [18] J.T. Schwartz, W-* Algebras, Gordon and Breach, New York, 1967. [19] J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics,

vol. 42, Springer-Verlang, New York, 1977. [20] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Graduate Texts

in Mathematics, vol. 102, Springer-Verlag, New York, 1984. [21] J. von Neumann and E.P. Wigner, Uber das Verhalten von Eigenwerten bei Adiabatischen

Prozessen, Physik Z. 30 (1929), 467-470; Engl, translation in "Symmetry in the Solid State" (R.S. Knox and A. Gold, eds.), W.A. Benjamin, New York, 1964, pp. 167-172.

261

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I N D E X

A2, 183 A4, 63 A5, 88 At, 187 abelian group, 1, 65 abelization of G, 42 act dually, 249 action, 3 A d ( » , 130 ad(X), 130 adjoint representation, 130 affine group, 10 algebraic integer, 43 algebraic number, 43 alternating space, 253 ambivalent, 48 anti-unitary, 30 automorphism, 2

B 2 , 183 B£i 187 basis, 206 branching relations, 108 Buckyball, 16 Burnside's theorem, 55

C, 13 Gn, 13 C/, 187 Cartan, 137 Cartan integers, 181, 197, 199, 201,

203 Cartan subalgebra, 166, 177, 197,

199, 200, 203 Cartan's criterion, 171 center, 39 central projection, 56

character, 35, 39, 157 character table, 42 class, ambivalent, 48 class function, 39 classical groups, 121, 137 Clebsch-Gordan integers, 29, 119, 222 Clifford algebra, 68, 152 Clifford group, 68-75 CL(n), 68 CL+(n), 73 column permutations, 101 commutant, 248 commutator subgroup, 35, 42 compact groups, 121 compact semisimple Lie group, 165 complex conjugate, 30 complex conjugate representation, 30-

34 complex representation, 47, 72, 74,

158, 242 conjugacy classes, 4, 39 convolution, 25, 38 coset, 3 cotangent bundle, 123 cotangent space, 123 covering group, 150 cube, 13, 15, 83 cycle, 9

D2n, 13 /#>(<,), 25 De, 187 degree, 21 differential equations, 124 differential form, 123 dihedral group, 8, 81 dimension theorem, 43

263

264 INDEX

direct product, 5, 7, 59 direct sum, 23 direct sum of irreps, 25 direct sum of matrix algebras, 38 direct sum representation, 24 disjoint, 56 disjoint cycles, 9, 95 dodecahedron, 13 dominant forms, 211 dominant weights, 211 double commutant theorem, 248 double cosets, 91 dual frame, 117 dual fundamental system, 185 dual group, 34 dual lattice, 210 Dynkin diagram, 186, 197, 199, 201,

203

E6, 187 E7, 187 Es, 187 edge, 192 equivalent, 56 Euclidean group, 10 exceptional Lie groups, 151 exterior product, 59

F4 , 187 /(.F), 97, 98, 107 face, 192 fibration, 150 finite abelian group, 65 Fourier analysis, 57 Fourier transforms, 58 Freudenthal, 223 Freudenthal's formula, 223 Frobenius, 77, 85, 89, 109 Frobenius character formula, 85, 109,

247 Frobenius reciprocity theorem, 89 fundamental group, 147 fundamental irreps, 228, 230, 231,

232 fundamental root, 184

fundamental system, 184, 193, 200, 203

fundamental translations, 181, 197, 199, 201, 203

fundamental weights, 211, 228, 230, 231, 232

G2, 183, 187 G-space, 2, 4 general linear, 21 generator, 167 GL(n,R), 135 group, 1 group, abelian, 1 group, topological, 2 group algebra, 25 group of quaternionic units, 31 group representation, 21 groups of order p2 , 19, 55 groups of order pq, 19, 55

Haar measure, 133 HausdorfT-Young inequality, 58 Hilbert space, 22 Hilbert-Schmidt inner product, 38 Hilbert-Schmidt operators, 257 homogeneous space, 4 homomorphism, 2 homotopy group, 146 homotopy product, 146

J, 13 J , 210, 228, 230, 231, 232 icosahedron, 13, 16 induced representations, 83-89 induction in stages, 90 infinitesimal Cartan-Stiefel diagram,

192 inner automorphism, 2, 3 inner product, 21 integral forms, 210 interior product, 59 intertwining map, 28 invariant subspace, 24 irreducible, 24 irreducible character, 35, 40

INDEX 265

irrep, 24 irreps of an abelian group, 28 isomorphism, 2 isotropy subgroup, 4

Jacobi identity, 125

/C, 208 Killing form, 170, 180, 197, 199, 201,

203 Klein, 11 Kostant, 222

Lp{G,dfi), 135 Lagrange's theorem, 3 lattice, 206 lattice of weights, 210 left coset, 3 left regular representation, 27 Lie algebra, 128 Lie bracket, 125 Lie group, 128 Lie-Trotter formula, 131 lowest form, 211

Mackey, 91 Mackey irreducibility criterion, 91 manifold, 122 matrix algebras, 38 matrix groups, 135 maximal root, 197, 199, 201, 203 maximal torus, 165, 196, 199, 200,

203 maximal weight, 217 minimal central projection, 56 minimal projection, 56 modulus, 133 multiplicity, 215

normal subgroup, 2 normalized, 184

octahedron, 13 one-dimensional representations, 42 orbit, 4 order, 3 orientable, 127

orientation, 127 oriented, 127 orthogonal matrices, 10 orthogonality relations, 36, 40, 157

p-group, 16-18 partitions, 96 Pauli a-matrices, 73 permutation group, 8 permutations, 95 Peter-Weyl theorem, 158, 257 7n(G,e), 147 7Tn(X,X0), 146 Platonic groups, 13 Poincare group, 10 point derivation, 122 projection, 56

quaternionic, 140 quaternionic representation, 30-34,

47, 50-54, 72, 74, 158, 244, 245

quotient, 5

K, 210, 228, 230, 231, 232 Racah, 222 rank, 166 real representation, 32, 47, 50-54,

72, 74, 158, 244, 245 regular element, 189 regular representation, 25 representations of abelian groups, 66-

67 representations of tori, 173 representations on quaternion vec­

tor spaces, 50-54 representations on real vector spaces,

50-54 restrictions, 59 right regular representation, 27 root elements, 181, 197, 199, 201,

204 roots, 178, 197, 199, 200, 203 root space, 178 root vectors, 178, 197, 199, 201, 203 rotations, 11

266 INDEX

row permutations, 101

53, 62 54, 83 55, 86 Sm , 250 5 n , 95, 109, 110, 253 Schur, 175 Schur's lemma, 27-28, 36 self-conjugate, 31 semidirect product, 6, 7, 77 semisimple, 225 semisimple Lie group, 165 sgn, 8 sign, 8 sign of permutation, 8 simple root, 184 simply connected, 147 singular element, 189 sl(2,C), 174 50(2n) , 137, 199, 230, 241, 245 50(2n + l) , 137 , 200, 231, 240, 245 50(3) , 145 SO(5), 145 SO{n), 11, 138, 150, 152 solvable, 225 Sp(l), 144 Sp(2), 145 Sp(n), 140, 142, 150, 203, 232, 239,

245 spin groups, 152 Spin(n), 151, 152 standard tableau, 97 Steinberg, 222 strongly dominant forms, 211 strongly dominant weights, 211 5(7(2), 144, 174 SU(n), 137, 150, 179, 182, 196, 228,

237, 242, 244, 250 subface, 192 subgroup, 2 subgroups of index 2, 60-62 Sylow theorems, 16 symmetric group, 95, 253 symmetric space, 253 symplectic group, 142

T, 13 T, 208 T o- T\ 102 tangent bundle, 123 tangent space, 123 tensor product, 29, 253 tensor product representation, 29 tetrahedral group, 82 tetrahedron, 13, 15 theorem of Probenius and Schur, 47,

256 topological group, 2, 121 torus, 165 transitive, 4

Ua, 190 unitarily equivalent, 22 unitary representation, 22, 26 universal covering group, 149

Vandermonde determinant, 238 Vandermonde function, 112 Vandermonde polynomial, 8 vector field, 123

weak convergence, 257 Wedderburn structure theorem, 39 weights, 210, 215 Weyl, 11, 78, 158, 174, 214, 221 Weyl chamber, 192 Weyl character formula, 217, 247 Weyl dimension formula, 221 Weyl group, 192, 197, 200, 201, 204,

215 Weyl integration formula, 213-215 Wigner, 77

y, 210 Young frame, 96 Young tableau, 97

Z, 206 Z(G), 190