Symmetry, representations, and invariants

Graduate Texts in Mathematics

255

Editorial Board
S. Axler
K.A. Ribet

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Roe Goodman
Nolan R. Wallach
Symmetry, Representations,
and Invariants

123


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Roe Goodman
Department of Mathematics
Rutgers University
Piscataway, NJ 08854-8019
USA


Nolan R. Wallach

Department of Mathematics
University of California, San Diego
La Jolla, CA 92093
USA


Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA


K.A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720-3840
USA


ISBN 978-0-387-79851-6
e-ISBN 978-0-387-79852-3
DOI 10.1007/978-0-387-79852-3
Springer Dordrecht Heidelberg London New York
Library of Congress Control Number: 2009927015
Mathematics Subject Classification (2000): 20G05, 14L35, 14M17, 17B10, 20C30, 20G20, 22E10,
22E46, 53B20, 53C35, 57M27
© Roe Goodman and Nolan R. Wallach 2009
Based on Representations and Invariants of the Classical Groups, Roe Goodman and Nolan R. Wallach,

Cambridge University Press, 1998, third corrected printing 2003.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)


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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Organization and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
1

Lie Groups and Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 General and Special Linear Groups . . . . . . . . . . . . . . . . . . . . .
1.1.2 Isometry Groups of Bilinear Forms . . . . . . . . . . . . . . . . . . . . .
1.1.3 Unitary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.4 Quaternionic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The Classical Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 General and Special Linear Lie Algebras . . . . . . . . . . . . . . . .

1.2.2 Lie Algebras Associated with Bilinear Forms . . . . . . . . . . . . .
1.2.3 Unitary Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.4 Quaternionic Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.5 Lie Algebras Associated with Classical Groups . . . . . . . . . . .
1.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Closed Subgroups of GL(n, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Lie Algebra of a Closed Subgroup of GL(n, R) . . . . . . . . . . .
1.3.4 Lie Algebras of the Classical Groups . . . . . . . . . . . . . . . . . . . .
1.3.5 Exponential Coordinates on Closed Subgroups . . . . . . . . . . .
1.3.6 Differentials of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . .
1.3.7 Lie Algebras and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Lie Algebra of an Algebraic Group . . . . . . . . . . . . . . . . . . . . .

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Contents

1.5

1.6

1.7

1.8
2

1.4.4 Algebraic Groups as Lie Groups . . . . . . . . . . . . . . . . . . . . . . . .
1.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rational Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Differential of a Rational Representation . . . . . . . . . . . . . . . . .
1.5.3 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Rational Representations of C . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.2 Rational Representations of C× . . . . . . . . . . . . . . . . . . . . . . . .
1.6.3 Jordan–Chevalley Decomposition . . . . . . . . . . . . . . . . . . . . . .
1.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Real Forms of Complex Algebraic Groups . . . . . . . . . . . . . . . . . . . . . .
1.7.1 Real Forms and Complex Conjugations . . . . . . . . . . . . . . . . . .
1.7.2 Real Forms of the Classical Groups . . . . . . . . . . . . . . . . . . . . .
1.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
45
47
47
49
53
54
55

55
57
58
61
62
62
65
67
68

Structure of Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1 Semisimple Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1.1 Toral Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.1.2 Maximal Torus in a Classical Group . . . . . . . . . . . . . . . . . . . . 72
2.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.2 Unipotent Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2.1 Low-Rank Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.2.2 Unipotent Generation of Classical Groups . . . . . . . . . . . . . . . 79
2.2.3 Connected Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3 Regular Representations of SL(2, C) . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.3.1 Irreducible Representations of sl(2, C) . . . . . . . . . . . . . . . . . . 84
2.3.2 Irreducible Regular Representations of SL(2, C) . . . . . . . . . . 86
2.3.3 Complete Reducibility of SL(2, C) . . . . . . . . . . . . . . . . . . . . . 88
2.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.4 The Adjoint Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.4.1 Roots with Respect to a Maximal Torus . . . . . . . . . . . . . . . . . 91
2.4.2 Commutation Relations of Root Spaces . . . . . . . . . . . . . . . . . . 95
2.4.3 Structure of Classical Root Systems . . . . . . . . . . . . . . . . . . . . . 99
2.4.4 Irreducibility of the Adjoint Representation . . . . . . . . . . . . . . 104

2.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2.5 Semisimple Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.5.1 Solvable Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2.5.2 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2.5.3 Geometry of Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.5.4 Conjugacy of Cartan Subalgebras . . . . . . . . . . . . . . . . . . . . . . . 122
2.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125


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2.6

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

3

Highest-Weight Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1 Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.1.1 Weyl Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.1.2 Root Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.1.3 Weight Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.1.4 Dominant Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.2 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.2.1 Theorem of the Highest Weight . . . . . . . . . . . . . . . . . . . . . . . . 148
3.2.2 Weights of Irreducible Representations . . . . . . . . . . . . . . . . . . 153

3.2.3 Lowest Weights and Dual Representations . . . . . . . . . . . . . . . 157
3.2.4 Symplectic and Orthogonal Representations . . . . . . . . . . . . . . 158
3.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.3 Reductivity of Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.3.1 Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.3.2 Casimir Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
3.3.3 Algebraic Proof of Complete Reducibility . . . . . . . . . . . . . . . 169
3.3.4 The Unitarian Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
3.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

4

Algebras and Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.1 Representations of Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . 175
4.1.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.1.2 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.1.3 Jacobson Density Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.1.4 Complete Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.1.5 Double Commutant Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.1.6 Isotypic Decomposition and Multiplicities . . . . . . . . . . . . . . . 184
4.1.7 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
4.2 Duality for Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.2.1 General Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.2.2 Products of Reductive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.2.3 Isotypic Decomposition of O[G] . . . . . . . . . . . . . . . . . . . . . . . . 199
4.2.4 Schur–Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.2.5 Commuting Algebra and Highest-Weight Vectors . . . . . . . . . 203
4.2.6 Abstract Capelli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

4.2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.3 Group Algebras of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.3.1 Structure of Group Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.3.2 Schur Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . 208
4.3.3 Fourier Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209


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4.3.4 The Algebra of Central Functions . . . . . . . . . . . . . . . . . . . . . . 211
4.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
4.4 Representations of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.4.1 Induced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.4.2 Characters of Induced Representations . . . . . . . . . . . . . . . . . . 217
4.4.3 Standard Representation of Sn . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.4.4 Representations of Sk on Tensors . . . . . . . . . . . . . . . . . . . . . . 221
4.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5

Classical Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.1 Polynomial Invariants for Reductive Groups . . . . . . . . . . . . . . . . . . . . 226
5.1.1 The Ring of Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5.1.2 Invariant Polynomials for Sn . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
5.2 Polynomial Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.2.1 First Fundamental Theorems for Classical Groups . . . . . . . . . 238

5.2.2 Proof of a Basic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
5.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3 Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.3.1 Tensor Invariants for GL(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5.3.2 Tensor Invariants for O(V ) and Sp(V ) . . . . . . . . . . . . . . . . . . . 248
5.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.4 Polynomial FFT for Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 256
5.4.1 Invariant Polynomials as Tensors . . . . . . . . . . . . . . . . . . . . . . . 256
5.4.2 Proof of Polynomial FFT for GL(V ) . . . . . . . . . . . . . . . . . . . . 258
5.4.3 Proof of Polynomial FFT for O(V ) and Sp(V ) . . . . . . . . . . . . 258
5.5 Irreducible Representations of Classical Groups . . . . . . . . . . . . . . . . . 259
5.5.1 Skew Duality for Classical Groups . . . . . . . . . . . . . . . . . . . . . . 259
5.5.2 Fundamental Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 268
5.5.3 Cartan Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
5.5.4 Irreducible Representations of GL(V ) . . . . . . . . . . . . . . . . . . . 273
5.5.5 Irreducible Representations of O(V ) . . . . . . . . . . . . . . . . . . . . 275
5.5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
5.6 Invariant Theory and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.6.1 Duality and the Weyl Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 278
5.6.2 GL(n)–GL(k) Schur–Weyl Duality . . . . . . . . . . . . . . . . . . . . . 282
5.6.3 O(n)–sp(k) Howe Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
5.6.4 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
5.6.5 Sp(n)–so(2k) Howe Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
5.6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
5.7 Further Applications of Invariant Theory . . . . . . . . . . . . . . . . . . . . . . 293
5.7.1 Capelli Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
5.7.2 Decomposition of S(S2 (V )) under GL(V ) . . . . . . . . . . . . . . . 295
5.7.3 Decomposition of S( 2 (V )) under GL(V ) . . . . . . . . . . . . . . . 296

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5.8

5.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

6

Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.1 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.1.1 Construction of Cliff (V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.1.2 Spaces of Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
6.1.3 Structure of Cliff (V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
6.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
6.2 Spin Representations of Orthogonal Lie Algebras . . . . . . . . . . . . . . . . 311
6.2.1 Embedding so(V ) in Cliff (V ) . . . . . . . . . . . . . . . . . . . . . . . . . . 312
6.2.2 Spin Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
6.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
6.3 Spin Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
6.3.1 Action of O(V ) on Cliff (V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
6.3.2 Algebraically Simply Connected Groups . . . . . . . . . . . . . . . . . 321
6.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
6.4 Real Forms of Spin(n, C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.4.1 Real Forms of Vector Spaces and Algebras . . . . . . . . . . . . . . . 323
6.4.2 Real Forms of Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . 325
6.4.3 Real Forms of Pin(n) and Spin(n) . . . . . . . . . . . . . . . . . . . . . . 325

6.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
6.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

7

Character Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.1 Character and Dimension Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.1.1 Weyl Character Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
7.1.2 Weyl Dimension Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
7.1.3 Commutant Character Formulas . . . . . . . . . . . . . . . . . . . . . . . . 337
7.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
7.2 Algebraic Group Approach to the Character Formula . . . . . . . . . . . . . 342
7.2.1 Symmetric and Skew-Symmetric Functions . . . . . . . . . . . . . . 342
7.2.2 Characters and Skew-Symmetric Functions . . . . . . . . . . . . . . 344
7.2.3 Characters and Invariant Functions . . . . . . . . . . . . . . . . . . . . . 346
7.2.4 Casimir Operator and Invariant Functions . . . . . . . . . . . . . . . . 347
7.2.5 Algebraic Proof of the Weyl Character Formula . . . . . . . . . . . 352
7.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
7.3 Compact Group Approach to the Character Formula . . . . . . . . . . . . . 354
7.3.1 Compact Form and Maximal Compact Torus . . . . . . . . . . . . . 354
7.3.2 Weyl Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
7.3.3 Fourier Expansions of Skew Functions . . . . . . . . . . . . . . . . . . 358
7.3.4 Analytic Proof of the Weyl Character Formula . . . . . . . . . . . . 360
7.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
7.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362


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8

Branching Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
8.1 Branching for Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
8.1.1 Statement of Branching Laws . . . . . . . . . . . . . . . . . . . . . . . . . . 364
8.1.2 Branching Patterns and Weight Multiplicities . . . . . . . . . . . . . 366
8.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
8.2 Branching Laws from Weyl Character Formula . . . . . . . . . . . . . . . . . . 370
8.2.1 Partition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
8.2.2 Kostant Multiplicity Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 371
8.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
8.3 Proofs of Classical Branching Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
8.3.1 Restriction from GL(n) to GL(n − 1) . . . . . . . . . . . . . . . . . . . 373
8.3.2 Restriction from Spin(2n + 1) to Spin(2n) . . . . . . . . . . . . . . . 375
8.3.3 Restriction from Spin(2n) to Spin(2n − 1) . . . . . . . . . . . . . . . 378
8.3.4 Restriction from Sp(n) to Sp(n − 1) . . . . . . . . . . . . . . . . . . . . 379
8.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9

Tensor Representations of GL(V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
9.1 Schur–Weyl Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
9.1.1 Duality between GL(n) and Sk . . . . . . . . . . . . . . . . . . . . . . . . 388
9.1.2 Characters of Sk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
9.1.3 Frobenius Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
9.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
9.2 Dual Reductive Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
9.2.1 Seesaw Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

9.2.2 Reciprocity Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
9.2.3 Schur–Weyl Duality and GL(k)–GL(n) Duality . . . . . . . . . . 405
9.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
9.3 Young Symmetrizers and Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . 407
9.3.1 Tableaux and Symmetrizers . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
9.3.2 Weyl Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
9.3.3 Standard Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
9.3.4 Projections onto Isotypic Components . . . . . . . . . . . . . . . . . . . 416
9.3.5 Littlewood–Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
9.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
9.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

10

Tensor Representations of O(V) and Sp(V) . . . . . . . . . . . . . . . . . . . . . . . 425
10.1 Commuting Algebras on Tensor Spaces . . . . . . . . . . . . . . . . . . . . . . . . 425
10.1.1 Centralizer Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
10.1.2 Generators and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
10.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
10.2 Decomposition of Harmonic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 435
10.2.1 Harmonic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
10.2.2 Harmonic Extreme Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
10.2.3 Decomposition of Harmonics for Sp(V ) . . . . . . . . . . . . . . . . . 440


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10.2.4 Decomposition of Harmonics for O(2l + 1) . . . . . . . . . . . . . . 442
10.2.5 Decomposition of Harmonics for O(2l) . . . . . . . . . . . . . . . . . 446
10.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
10.3 Riemannian Curvature Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
10.3.1 The Space of Curvature Tensors . . . . . . . . . . . . . . . . . . . . . . . . 453
10.3.2 Orthogonal Decomposition of Curvature Tensors . . . . . . . . . . 455
10.3.3 The Space of Weyl Curvature Tensors . . . . . . . . . . . . . . . . . . . 458
10.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
10.4 Invariant Theory and Knot Polynomials . . . . . . . . . . . . . . . . . . . . . . . . 461
10.4.1 The Braid Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
10.4.2 Orthogonal Invariants and the Yang–Baxter Equation . . . . . . 463
10.4.3 The Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
10.4.4 The Jones Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
10.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
10.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
11

Algebraic Groups and Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . 479
11.1 General Properties of Linear Algebraic Groups . . . . . . . . . . . . . . . . . . 479
11.1.1 Algebraic Groups as Affine Varieties . . . . . . . . . . . . . . . . . . . . 479
11.1.2 Subgroups and Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 481
11.1.3 Group Structures on Affine Varieties . . . . . . . . . . . . . . . . . . . . 484
11.1.4 Quotient Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
11.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
11.2 Structure of Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
11.2.1 Commutative Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . 491
11.2.2 Unipotent Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
11.2.3 Connected Algebraic Groups and Lie Groups . . . . . . . . . . . . . 496
11.2.4 Simply Connected Semisimple Groups . . . . . . . . . . . . . . . . . . 497
11.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

11.3 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
11.3.1 G-Spaces and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
11.3.2 Flag Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
11.3.3 Involutions and Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . 506
11.3.4 Involutions of Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . 507
11.3.5 Classical Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 510
11.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
11.4 Borel Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
11.4.1 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
11.4.2 Lie–Kolchin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
11.4.3 Structure of Connected Solvable Groups . . . . . . . . . . . . . . . . . 522
11.4.4 Conjugacy of Borel Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . 524
11.4.5 Centralizer of a Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
11.4.6 Weyl Group and Regular Semisimple Conjugacy Classes . . . 526
11.4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530
11.5 Further Properties of Real Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531


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11.5.1 Groups with a Compact Real Form . . . . . . . . . . . . . . . . . . . . . 531
11.5.2 Polar Decomposition by a Compact Form . . . . . . . . . . . . . . . . 536
11.6 Gauss Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
11.6.1 Gauss Decomposition of GL(n, C) . . . . . . . . . . . . . . . . . . . . . 538
11.6.2 Gauss Decomposition of an Algebraic Group . . . . . . . . . . . . . 540
11.6.3 Gauss Decomposition for Real Forms . . . . . . . . . . . . . . . . . . . 541
11.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
12

Representations on Spaces of Regular Functions . . . . . . . . . . . . . . . . . . 545
12.1 Some General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
12.1.1 Isotypic Decomposition of O[X] . . . . . . . . . . . . . . . . . . . . . . . . 546
12.1.2 Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
12.1.3 Function Models for Irreducible Representations . . . . . . . . . . 549
12.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
12.2 Multiplicity-Free Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
12.2.1 Multiplicities and B-Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
12.2.2 B-Eigenfunctions for Linear Actions . . . . . . . . . . . . . . . . . . . . 553
12.2.3 Branching from GL(n) to GL(n − 1) . . . . . . . . . . . . . . . . . . . . 554
12.2.4 Second Fundamental Theorems . . . . . . . . . . . . . . . . . . . . . . . . 557
12.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
12.3 Regular Functions on Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . 566
12.3.1 Iwasawa Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
12.3.2 Examples of Iwasawa Decompositions . . . . . . . . . . . . . . . . . . 575
12.3.3 Spherical Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
12.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
12.4 Isotropy Representations of Symmetric Spaces . . . . . . . . . . . . . . . . . . 588
12.4.1 A Theorem of Kostant and Rallis . . . . . . . . . . . . . . . . . . . . . . . 589
12.4.2 Invariant Theory of Reflection Groups . . . . . . . . . . . . . . . . . . . 590
12.4.3 Classical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
12.4.4 Some Results from Algebraic Geometry . . . . . . . . . . . . . . . . . 597
12.4.5 Proof of the Kostant–Rallis Theorem . . . . . . . . . . . . . . . . . . . . 601
12.4.6 Some Remarks on the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 605
12.4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

A

Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
A.1 Affine Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
A.1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
A.1.2 Zariski Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
A.1.3 Products of Affine Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
A.1.4 Principal Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
A.1.5 Irreducible Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
A.1.6 Transcendence Degree and Dimension . . . . . . . . . . . . . . . . . . 619
A.1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621


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A.2 Maps of Algebraic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
A.2.1 Rational Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
A.2.2 Extensions of Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 623
A.2.3 Image of a Dominant Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
A.2.4 Factorization of a Regular Map . . . . . . . . . . . . . . . . . . . . . . . . . 626
A.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
A.3 Tangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
A.3.1 Tangent Space and Differentials of Maps . . . . . . . . . . . . . . . . 628
A.3.2 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
A.3.3 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
A.3.4 Differential Criterion for Dominance . . . . . . . . . . . . . . . . . . . . 632
A.4 Projective and Quasiprojective Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

A.4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
A.4.2 Products of Projective Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
A.4.3 Regular Functions and Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
B

Linear and Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
B.1 Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
B.1.1 Primary Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
B.1.2 Additive Jordan Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 644
B.1.3 Multiplicative Jordan Decomposition . . . . . . . . . . . . . . . . . . . 645
B.2 Multilinear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
B.2.1 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
B.2.2 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
B.2.3 Symmetric Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
B.2.4 Alternating Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
B.2.5 Determinants and Gauss Decomposition . . . . . . . . . . . . . . . . . 654
B.2.6 Pfaffians and Skew-Symmetric Matrices . . . . . . . . . . . . . . . . . 656
B.2.7 Irreducibility of Determinants and Pfaffians . . . . . . . . . . . . . . 659

C

Associative Algebras and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
C.1 Some Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
C.1.1 Filtered and Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 661
C.1.2 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
C.1.3 Symmetric Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
C.1.4 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
C.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
C.2 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668
C.2.1 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668

C.2.2 Poincar´e–Birkhoff–Witt Theorem . . . . . . . . . . . . . . . . . . . . . . 670
C.2.3 Adjoint Representation of Enveloping Algebra . . . . . . . . . . . . 672


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xiv

D

Contents

Manifolds and Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
D.1 C∞ Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
D.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
D.1.2 Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
D.1.3 Differential Forms and Integration . . . . . . . . . . . . . . . . . . . . . . 683
D.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686
D.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
D.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
D.2.2 Lie Algebra of a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
D.2.3 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
D.2.4 Integration on Lie Groups and Homogeneous Spaces . . . . . . 692
D.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709


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Preface

Symmetry, in the title of this book, should be understood as the geometry of Lie
(and algebraic) group actions. The basic algebraic and analytic tools in the study
of symmetry are representation and invariant theory. These three threads are precisely the topics of this book. The earlier chapters can be studied at several levels. An advanced undergraduate or beginning graduate student can learn the theory
for the classical groups using only linear algebra, elementary abstract algebra, and
advanced calculus, with further exploration of the key examples and concepts in
the numerous exercises following each section. The more sophisticated reader can
progress through the first ten chapters with occasional forward references to Chapter 11 for general results about algebraic groups. This allows great flexibility in the
use of this book as a course text. The authors have used various chapters in a variety
of courses; we suggest ways in which courses can be based on the book later in this
preface. Finally, we have taken care to make the main theorems and applications
meaningful for the reader who wishes to use the book as a reference to this vast
subject.
The authors are gratified that their earlier text, Representations and Invariants of
the Classical Groups [56], was well received. The present book has the same aim: an
entry into the powerful techniques of Lie and algebraic group theory. The parts of the
previous book that have withstood the authors’ many revisions as they lectured from
its material have been retained; these parts appear here after substantial rewriting
and reorganization. The first four chapters are, in large part, newly written and offer
a more direct and elementary approach to the subject. Several of the later parts of
the book are also new. While we continue to look upon the classical groups as both
fundamental in their own right and as important examples for the general theory, the
results are now stated and proved in their natural generality. These changes justify
the more accurate new title for the present book.
We have taken special care to make the book readable at many levels of detail.
A reader desiring only the statement of a pertinent result can find it through the
table of contents and index, and then read and study it through the examples of its
use that are generally given. A more serious reader wishing to delve into a proof of

the result can read in detail a more computational proof that uses special properties

xv


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xvi

Preface

of the classical groups, or, perhaps in a second reading, the proof in the general
case (with occasional forward references to results from later chapters). Usually,
there is a third possibility of a proof using analytic methods. Some material in the
earlier book, although important in its own right, has been eliminated or replaced.
There are new proofs of some of the key results of the theory such as the theorem
of the highest weight, the theorem on complete reducibility, the duality theorem,
and the Weyl character formula. We hope that our new presentation will make these
fundamental tools more accessible.
The last two chapters of the book develop, via a basic introduction to complex
algebraic groups, what has come to be called geometric invariant theory. This includes the notion of quotient space and the representation-theoretic analysis of the
regular functions on a space with an algebraic group action. A full description of the
material covered in the book is given later in the preface.
When our earlier text appeared there were few other introductions to the area.
The most prominent included the fundamental text of Hermann Weyl, The Classical
Groups: Their Invariants and Representations [164] and Chevalley’s The Theory of
Lie groups I [33], together with the more recent text Lie Algebras by Humphreys
[76]. These remarkable volumes should be on the bookshelf of any serious student of
the subject. In the interim, several other texts have appeared that cover, for the most
part, the material in Chevalley’s classic with extensions of his analytic group theory
to Lie group theory and that also incorporate much of the material in Humphrey’s

text. Two books with a more substantial overlap but philosophically very different
from ours are those by Knapp [86] and Procesi [123]. There is much for a student
to learn from both of these books, which give an exposition of Weyl’s methods in
invariant theory that is different in emphasis from our book. We have developed
the combinatorial aspects of the subject as consequences of the representations and
invariants of the classical groups. In Hermann Weyl (and the book of Procesi) the
opposite route is followed: the representations and invariants of the classical groups
rest on a combinatorial determination of the representations of the symmetric group.
Knapp’s book is more oriented toward Lie group theory.

Organization
The logical organization of the book is illustrated in the chapter and section dependency chart at the end of the preface. A chapter or section listed in the chart depends
on the chapters to which it is connected by a horizontal or rising line. This chart has
a central spine; to the right are the more geometric aspects of the subject and on the
left the more algebraic aspects. There are several intermediate terminal nodes in this
chart (such as Sections 5.6 and 5.7, Chapter 6, and Chapters 9–10) that can serve as
goals for courses or self study.
Chapter 1 gives an elementary approach to the classical groups, viewed either as
Lie groups or algebraic groups, without using any deep results from differentiable
manifold theory or algebraic geometry. Chapter 2 develops the basic structure of
the classical groups and their Lie algebras, taking advantage of the defining representations. The complete reducibility of representations of sl(2, C) is established by
a variant of Cartan’s original proof. The key Lie algebra results (Cartan subalge-


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Preface

xvii

bras and root space decomposition) are then extended to arbitrary semisimple Lie

algebras.
Chapter 3 is devoted to Cartan’s highest-weight theory and the Weyl group. We
give a new algebraic proof of complete reducibility for semisimple Lie algebras
following an argument of V. Kac; the only tools needed are the complete reducibility
for sl(2, C) and the Casimir operator. The general treatment of associative algebras
and their representations occurs in Chapter 4, where the key result is the general
duality theorem for locally regular representations of a reductive algebraic group.
The unifying role of the duality theorem is even more prominent throughout the
book than it was in our previous book.
The machinery of Chapters 1–4 is then applied in Chapter 5 to obtain the principal results in classical representations and invariant theory: the first fundamental
theorems for the classical groups and the application of invariant theory to representation theory via the duality theorem.
Chapters 6, on spinors, follows the corresponding chapter from our previous
book, with some corrections and additional exercises. For the main result in Chapter 7—the Weyl character formula—we give a new algebraic group proof using the
radial component of the Casimir operator (replacing the proof via Lie algebra cohomology in the previous book). This proof is a differential operator analogue of
Weyl’s original proof using compact real forms and the integration formula, which
we also present in detail. The treatment of branching laws in Chapter 8 follows the
same approach (due to Kostant) as in the previous book.
Chapters 9–10 apply all the machinery developed in previous chapters to analyze
the tensor representations of the classical groups. In Chapter 9 we have added a
discussion of the Littlewood–Richardson rule (including the role of the GL(n, C)
branching law to reduce the proof to a well-known combinatorial construction). We
have removed the partial harmonic decomposition of tensor space under orthogonal
and symplectic groups that was treated in Chapter 10 of the previous book, and
replaced it with a representation-theoretic treatment of the symmetry properties of
curvature tensors for pseudo-Riemannian manifolds.
The general study of algebraic groups over C and homogeneous spaces begins
in Chapter 11 (with the necessary background material from algebraic geometry in
Appendix A). In Lie theory the examples are, in many cases, more difficult than the
general theorems. As in our previous book, every new concept is detailed with its
meaning for each of the classical groups. For example, in Chapter 11 every classical symmetric pair is described and a model is given for the corresponding affine

variety, and in Chapter 12 the (complexified) Iwasawa decomposition is worked out
explicitly. Also in Chapter 12 a proof of the celebrated Kostant–Rallis theorem for
symmetric spaces is given and every implication for the invariant theory of classical
groups is explained.
This book can serve for several different courses. An introductory one-term
course in Lie groups, algebraic groups, and representation theory with emphasis
on the classical groups can be based on Chapters 1–3 (with reference to Appendix
D as needed). Chapters 1–3 and 11 (with reference to Appendix A as needed) can
be the core of a one-term introductory course on algebraic groups in characteris-


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xviii

Preface

tic zero. For students who have already had an introductory course in Lie algebras
and Lie groups, Chapters 3 and 4 together with Chapters 6–10 contain ample material for a second course emphasizing representations, character formulas, and their
applications. An alternative (more advanced) second-term course emphasizing the
geometric side of the subject can be based on topics from Chapters 3, 4, 11, and 12.
A year-long course on representations and classical invariant theory along the lines
of Weyl’s book would follow Chapters 1–5, 7, 9, and 10. The exercises have been
revised and many new ones added (there are now more than 350, most with several
parts and detailed hints for solution). Although none of the exercises are used in
the proofs of the results in the book, we consider them an essential part of courses
based on this book. Working through a significant number of the exercises helps a
student learn the general concepts, fine structure, and applications of representation
and invariant theory.

Acknowledgments

In the end-of-chapter notes we have attempted to give credits for the results in the
book and some idea of the historical development of the subject. We apologize to
those whose works we have neglected to cite and for incorrect attributions. We are
indebted to many people for finding errors and misprints in the many versions of
the material in this book and for suggesting ways to improve the exposition. In
particular we would like to thank Ilka Agricola, Laura Barberis, Bachir Bekka, Enriqueta Rodr´ıguez Carrington, Friedrich Knop, Hanspeter Kraft, Peter Landweber,
and Tomasz Przebinda. Chapters of the book have been used in many courses, and
the interaction with the students was very helpful in arriving at the final version. We
thank them all for their patience, comments, and sharp eyes. During the first year
that we were writing our previous book (1989–1990), Roger Howe gave a course
at Rutgers University on basic invariant theory. We thank him for many interesting
conversations on this subject.
The first-named author is grateful for sabbatical leaves from Rutgers University
and visiting appointments at the University of California, San Diego; the Universit´e
de Metz; Hong Kong University; and the National University of Singapore that were
devoted to this book. He thanks the colleagues who arranged programs in which he
lectured on this material, including Andrzei Hulanicki and Ewa Damek (University
of Wrocław), Bachir Bekka and Jean Ludwig (Universit´e de Metz), Ngai-Ming Mok
(Hong Kong University), and Eng-Chye Tan and Chen-Bo Zhu (National University
of Singapore). He also is indebted to the mathematics department of the University
of California, Berkeley, for its hospitality during several summers of writing the
book. The second-named author would like to thank the University of California,
San Diego, for sabbatical leaves that were dedicated to this project, and to thank
the National Science Foundation for summer support. We thank David Kramer for a
superb job of copyediting, and we are especially grateful to our editor Ann Kostant,
who has steadfastly supported our efforts on the book from inception to publication.
New Brunswick, New Jersey
San Diego, California
February 2009

Roe Goodman
Nolan R. Wallach


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Organization and Notation
B. Linear and
Multilinear Algebra

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

Chap. 1 Classical Groups,
Lie Groups, and Algebraic Groups

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

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

D. Manifolds and
Lie Groups

Chap. 2 Structure of Classical Groups and Semisimple Lie Algebras
...
...

..
..

C. Associative and
Lie Algebras

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

Chap. 3 Highest-Weight Theory
...
...
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...
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§4.1 Representations of Algebras
§4.2 Duality for Group Representations
...
...
...
..

§4.3 Group Algebras of Finite Groups
§4.4 Representations of Finite Groups
...
...
...
...
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...
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...
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...
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.

§5.1–5.4 Classical Invariant Theory
...
...
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.....

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

§5.5 Irreducible Representations of
Classical Groups

§5.6–5.7 Applications of
Invariant Theory and Duality
Chap. 6 Spinors

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

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

Chap. 7 Character Formulas
...
..
...
.

Chap. 8 Branching Laws
..
...
...
..

Chaps. 9, 10 Tensor Representations
of GL(V ), O(V ), and Sp(V )

A. Algebraic
Geometry
...
...
...
...
...
..

§11.1 Structure of Algebraic Groups
§11.2 Homogeneous Spaces
§11.3 Borel Subgroups

...
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§11.4 Properties of Real Forms
§11.5 Gauss Decomposition

§12.1 Representations on Function Spaces
...
...
...
...

...
...
...

§12.4 Isotropy Representations
of Symmetric Spaces

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

§12.2 Multiplicity-Free Spaces
...
..
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.

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

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§12.3 Representations on Symmetric Spaces

Dependency Chart among Chapters and Sections
“O,” said Maggie, pouting, “I dare say I could make it out, if I’d learned what goes before,
as you have.” “But that’s what you just couldn’t, Miss Wisdom,” said Tom. “For it’s all
the harder when you know what goes before: for then you’ve got to say what Definition 3.
is and what Axiom V. is.”
George Eliot, The Mill on the Floss

xix


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xx

Organization and Notation

Some Standard Notation
#S number of elements in set S (also denoted by Card(S) and |S|)

δi j Kronecker delta (1 if i = j, 0 otherwise)

N, Z, Q nonnegative integers, integers, rational numbers
R, C, H real numbers, complex numbers, quaternions
C× multiplicative group of nonzero complex numbers
[x] greatest integer ≤ x if x is real

Fn n × 1 column vectors with entries in field F

Mk,n k × n complex matrices (Mn when k = n)
Mn (F) n × n matrices with entries in field F

GL(n, F) invertible n × n matrices with entries from field F
In n × n identity matrix (or I when n understood)
dimV dimension of a vector space V
V ∗ dual space to vector space V
v∗ , v natural duality pairing between V ∗ and V
Span(S) linear span of subset S in a vector space.
End(V ) linear transformations on vector space V
GL(V ) invertible linear transformations on vector space V
tr(A) trace of square matrix A
det(A) determinant of square matrix A
At transpose of matrix A
A∗ conjugate transpose of matrix A
diag[a1 , . . . , an ] diagonal matrix
Vi direct sum of vector spaces Vi
k

V k-fold tensor product of vector space V (also denoted by V ⊗k )

Sk (V ) k-fold symmetric tensor product of vector space V
k

(V ) k-fold skew-symmetric tensor product of vector space V

O[X] regular functions on algebraic set X
Other notation is generally defined at its first occurrence and appears in the index of
notation at the end of the book.


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Chapter 1

Lie Groups and Algebraic Groups

Abstract Hermann Weyl, in his famous book The Classical Groups, Their Invariants and Representations [164], coined the name classical groups for certain
families of matrix groups. In this chapter we introduce these groups and develop
the basic ideas of Lie groups, Lie algebras, and linear algebraic groups. We show
how to put a Lie group structure on a closed subgroup of the general linear group
and determine the Lie algebras of the classical groups. We develop the theory of
complex linear algebraic groups far enough to obtain the basic results on their Lie
algebras, rational representations, and Jordan–Chevalley decompositions (we defer
the deeper results about algebraic groups to Chapter 11). We show that linear algebraic groups are Lie groups, introduce the notion of a real form of an algebraic
group (considered as a Lie group), and show how the classical groups introduced at
the beginning of the chapter appear as real forms of linear algebraic groups.

1.1 The Classical Groups
The classical groups are the groups of invertible linear transformations of finitedimensional vector spaces over the real, complex, and quaternion fields, together
with the subgroups that preserve a volume form, a bilinear form, or a sesquilinear
form (the forms being nondegenerate and symmetric or skew-symmetric).

1.1.1 General and Special Linear Groups
Let F denote either the field of real numbers R or the field of complex numbers
C, and let V be a finite-dimensional vector space over F. The set of invertible linear transformations from V to V will be denoted by GL(V ). This set has a group
structure under composition of transformations, with identity element the identity
transformation I(x) = x for all x ∈ V . The group GL(V ) is the first of the classical
R. Goodman, N.R. Wallach, Symmetry, Representations, and Invariants,
Graduate Texts in Mathematics 255, DOI 10.1007/978-0-387-79852-3_1,
© Roe Goodman and Nolan R. Wallach 2009

1


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2

1 Lie Groups and Algebraic Groups

groups. To study it in more detail, we recall some standard terminology related to
linear transformations and their matrices.
Let V and W be finite-dimensional vector spaces over F. Let {v1 , . . . , vn } and
/ W is a linear map
{w1 , . . . , wm } be bases for V and W , respectively. If T : V
then
m

T v j = ∑ ai j wi

for j = 1, . . . , n

i=1

with ai j ∈ F. The numbers ai j are called the matrix coefficients or entries of T with
respect to the two bases, and the m × n array


a11 a12 · · · a1n
 a21 a22 · · · a2n 



A= .
.. . . .. 
 ..
. . 
.
am1 am2 · · · amn
is the matrix of T with respect to the two bases. When the elements of V and W are
identified with column vectors in Fn and Fm using the given bases, then action of T
becomes multiplication by the matrix A.
/ U be another linear transformation, with U an l-dimensional vecLet S : W
tor space with basis {u1 , . . . , ul }, and let B be the matrix of S with respect to the
bases {w1 , . . . , wm } and {u1 , . . . , ul }. Then the matrix of S ◦ T with respect to the
bases {v1 , . . . , vn } and {u1 , . . . , ul } is given by BA, with the product being the usual
product of matrices.
We denote the space of all n × n matrices over F by Mn (F), and we denote the
n × n identity matrix by I (or In if the size of the matrix needs to be indicated);
it has entries δi j = 1 if i = j and 0 otherwise. Let V be an n-dimensional vector
/ V is a linear map we write µ(T )
space over F with basis {v1 , . . . , vn }. If T : V
for the matrix of T with respect to this basis. If T, S ∈ GL(V ) then the preceding
observations imply that µ(S ◦ T ) = µ(S)µ(T ). Furthermore, if T ∈ GL(V ) then
µ(T ◦ T −1 ) = µ(T −1 ◦ T ) = µ(Id) = I. The matrix A ∈ Mn (F) is said to be invertible
if there is a matrix B ∈ Mn (F) such that AB = BA = I. We note that a linear map
/ V is in GL(V ) if and only if its matrix µ(T ) is invertible. We also recall
T :V
that a matrix A ∈ Mn (F) is invertible if and only if its determinant is nonzero.
We will use the notation GL(n, F) for the set of n × n invertible matrices with
coefficients in F. Under matrix multiplication GL(n, F) is a group with the identity
matrix as identity element. We note that if V is an n-dimensional vector space over
/ GL(n, F) corresponding to

F with basis {v1 , . . . , vn }, then the map µ : GL(V )
this basis is a group isomorphism. The group GL(n, F) is called the general linear
group of rank n.
If {w1 , . . . , wn } is another basis of V , then there is a matrix g ∈ GL(n, F) such
that
n

w j = ∑ gi j vi
i=1

n

and v j = ∑ hi j wi
i=1

for j = 1, . . . , n ,


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1.1 The Classical Groups

3

with [hi j ] the inverse matrix to [gi j ]. Suppose that T is a linear transformation from
V to V , that A = [ai j ] is the matrix of T with respect to a basis {v1 , . . . , vn }, and that
B = [bi j ] is the matrix of T with respect to another basis {w1 , . . . , wn }. Then
Tw j = T

∑ gi j vi
i

= ∑ gi j
i

= ∑ gi j T vi

∑ aki vk
k

i

=∑
l

∑ ∑ hlk aki gi j
k

wl

i

for j = 1, . . . , n. Thus B = g−1 Ag is similar to the matrix A.

Special Linear Group
The special linear group SL(n, F) is the set of all elements A of Mn (F) such that
det(A) = 1. Since det(AB) = det(A) det(B) and det(I) = 1, we see that the special
linear group is a subgroup of GL(n, F).
We note that if V is an n-dimensional vector space over F with basis {v1 , . . . , vn }
/ GL(n, F) is the map previously defined, then the group
and if µ : GL(V )

µ −1 (SL(n, F)) = {T ∈ GL(V ) : det(µ(T )) = 1}
is independent of the choice of basis, by the change of basis formula. We denote this
group by SL(V ).

1.1.2 Isometry Groups of Bilinear Forms
/F
Let V be an n-dimensional vector space over F. A bilinear map B : V × V
is called a bilinear form. We denote by O(V, B) (or O(B) when V is understood)
the set of all g ∈ GL(V ) such that B(gv, gw) = B(v, w) for all v, w ∈ V . We note that
O(V, B) is a subgroup of GL(V ); it is called the isometry group of the form B.
Let {v1 , . . . , vn } be a basis of V and let Γ ∈ Mn (F) be the matrix with Γi j =
B(vi , v j ). If g ∈ GL(V ) has matrix A = [ai j ] relative to this basis, then
B(gvi , gv j ) = ∑ aki al j B(vk , vl ) = ∑ akiΓkl al j .
k,l

k,l

Thus if At denotes the transposed matrix [ci j ] with ci j = a ji , then the condition that
g ∈ O(B) is that
Γ = At Γ A .
(1.1)
Recall that a bilinear form B is nondegenerate if B(v, w) = 0 for all w implies
that v = 0, and likewise B(v, w) = 0 for all v implies that w = 0. In this case we
/ V is linear and satisfies
have detΓ = 0. Suppose B is nondegenerate. If T : V


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4

1 Lie Groups and Algebraic Groups

B(T v, Tw) = B(v, w) for all v, w ∈ V , then det(T ) = 0 by formula (1.1). Hence T ∈
O(B). The next two subsections will discuss the most important special cases of this
class of groups.

Orthogonal Groups
We start by introducing the matrix groups; later we will identify these groups with
isometry groups of certain classes of bilinear forms. Let O(n, F) denote the set of
all g ∈ GL(n, F) such that ggt = I. That is, gt = g−1 . We note that (AB)t = Bt At and
if A, B ∈ GL(n, F) then (AB)−1 = B−1 A−1 . It is therefore obvious that O(n, F) is a
subgroup of GL(n, F). This group is called the orthogonal group of n × n matrices
over F. If F = R we introduce the indefinite orthogonal groups, O(p, q), with p+q =
n and p, q ∈ N. Let
I 0
I p, q = p
0 −Iq
and define
O(p, q) = {g ∈ Mn (R) : gt I p,q g = I p,q } .

We note that O(n, 0) = O(0, n) = O(n, R). Also, if


0 0 ··· 1
 .. .. . . .. 


s=. . . .
 0 1 ··· 0 
1 0 ··· 0

is the matrix with entries 1 on the skew diagonal ( j = n + 1 − i) and all other entries
0, then s = s−1 = st and sI p,q s−1 = sI p,q s = sI p,q s = −Iq,p . Thus the map
ϕ : O(p, q)

/ GL(n, R)

given by ϕ(g) = sgs defines an isomorphism of O(p, q) onto O(q, p).
We will now describe these groups in terms of bilinear forms.
Definition 1.1.1. Let V be a vector space over R and let M be a symmetric bilinear
form on V . The form M is positive definite if M(v, v) > 0 for every v ∈ V with v = 0.
Lemma 1.1.2. Let V be an n-dimensional vector space over F and let B be a symmetric nondegenerate bilinear form over F.
1. If F = C then there exists a basis {v1 , . . . , vn } of V such that B(vi , v j ) = δi j .
2. If F = R then there exist integers p, q ≥ 0 with p + q = n and a basis {v1 , . . . , vn }
of V such that B(vi , v j ) = εi δi j with εi = 1 for i ≤ p and εi = −1 for i > p. Furthermore, if we have another such basis then the corresponding integers (p, q)
are the same.


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1.1 The Classical Groups

5

Remark 1.1.3. The basis for V in part (2) is called a pseudo-orthonormal basis relative to B, and the number p − q is called the signature of the form (we will also
call the pair (p, q) the signature of B). A form is positive definite if and only if its
signature is n. In this case a pseudo-orthonormal basis is an orthonormal basis in the
usual sense.
Proof. We first observe that if M is a symmetric bilinear form on V such that
M(v, v) = 0 for all v ∈ V , then M = 0. Indeed, using the symmetry and bilinearity we have
4M(v, w) = M(v + w, v + w) − M(v − w, v − w) = 0

for all v, w ∈ V .

(1.2)

We now construct a basis {w1 , . . . , wn } of V such that
B(wi , w j ) = 0

for i = j and

B(wi , wi ) = 0

(such a basis is called an orthogonal basis with respect to B). The argument is
by induction on n. Since B is nondegenerate, there exists a vector wn ∈ V with
B(wn , wn ) = 0 by (1.2). If n = 1 we are done. If n > 1, set
V = {v ∈ V : B(wn , v) = 0} .
For v ∈ V set

v = v−

B(v, wn )
wn .
B(wn , wn )

Clearly, v ∈ V ; hence V = V + Fwn . In particular, this shows that dimV = n − 1.
We assert that the form B = B|V ×V is nondegenerate on V . Indeed, if v ∈ V satisfies B(v , w) = 0 for all w ∈ V , then B(v , w) = 0 for all w ∈ V , since B(v , wn ) = 0.
Hence v = 0, proving nondegeneracy of B . We may assume by induction that there
exists a B -orthogonal basis {w1 , . . . , wn−1 } for V . Then it is clear that {w1 , . . . , wn }
is a B-orthogonal basis for V .
If F = C let {w1 , . . . , wn } be an orthogonal basis of V with respect to B and let

zi ∈ C be a choice of square root of B(wi , wi ). Setting vi = (zi )−1 wi , we then obtain
the desired normalization B(vi , v j ) = δi j .
Now let F = R. We rearrange the indices (if necessary) so that B(wi , wi ) ≥
B(wi+1 , wi+1 ) for i = 1, . . . , n − 1. Let p = 0 if B(w1 , w1 ) < 0. Otherwise, let
p = max{i : B(wi , wi ) > 0} .
Then B(wi , wi ) < 0 for i > p. Take zi to be a square root of B(wi , wi ) for i ≤ p, and
take zi to be a square root of −B(wi , wi ) for i > p. Setting vi = (zi )−1 wi , we now
have B(vi , v j ) = εi δi j .
We are left with proving that the integer p is intrinsic to B. Take any basis
{v1 , . . . , vn } such that B(vi , v j ) = εi δi j with εi = 1 for i ≤ p and εi = −1 for i > p.
Set
V+ = Span{v1 , . . . , v p } ,
V− = Span{v p+1 , . . . , vn } .


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6

1 Lie Groups and Algebraic Groups

/ V+ be the projection onto the first
Then V = V+ ⊕ V− (direct sum). Let π : V
factor. We note that B|V+ ×V+ is positive definite. Let W be any subspace of V such
that B|W ×W is positive definite. Suppose that w ∈ W and π(w) = 0. Then w ∈ V− , so
it can be written as w = ∑i>p ai vi . Hence
B(w, w) =

∑

i, j>p

ai a j B(vi , v j ) = − ∑ a2i ≤ 0 .
i>p

Since B|W ×W has been assumed to be positive definite, it follows that w = 0. This
/ V+ is injective, and hence dimW ≤ dimV+ = p. Thus p
implies that π : W
is uniquely determined as the maximum dimension of a subspace on which B is
positive definite.
The following result follows immediately from Lemma 1.1.2.
Proposition 1.1.4. Let B be a nondegenerate symmetric bilinear form on an ndimensional vector space V over F.
1. Let F = C. If {v1 , . . . , vn } is an orthonormal basis for V with respect to B, then
/ O(n, F) defines a group isomorphism.
µ : O(V, B)
2. Let F = R. If B has signature (p, n − p) and {v1 , . . . , vn } is a pseudo-orthonormal
/ O(p, n − p) is a group isomorphism.
basis of V , then µ : O(V, B)

Here µ(g), for g ∈ GL(V ), is the matrix of g with respect to the given basis.
The special orthogonal group over F is the subgroup
SO(n, F) = O(n, F) ∩ SL(n, F)
of O(n, F). The indefinite special orthogonal groups are the groups
SO(p, q) = O(p, q) ∩ SL(p + q, R) .

Symplectic Group
0 I with I the n × n identity matrix. The symplectic group of rank n
We set J = −I
0
over F is defined to be

Sp(n, F) = {g ∈ M2n (F) : gt Jg = J}.
As in the case of the orthogonal groups one sees without difficulty that Sp(n, F) is a
subgroup of GL(2n, F).
We will now look at the coordinate-free version of these groups. A bilinear form
B is called skew-symmetric if B(v, w) = −B(w, v). If B is skew-symmetric and nondegenerate, then m = dimV must be even, since the matrix of B relative to any basis
for V is skew-symmetric and has nonzero determinant.