To an algebraist the theory of group characters presents one of those fascinating situations, where the structure of an abstract system is elucidated by a unique set of numbers inherent in the system. But the subject also has a practical aspect, since group characters have gained importance in several branches of science, in which considerations of symmetry play a decisive part. This is an introductory text, suitable for final-year undergraduates or postgraduate students. The only prerequisites are a standard knowledge of linear algebra and a modest acquaintance with group theory. Especial care has been taken to explain how group characters are computed. The character tables of most of the familiar accessible groups are either constructed in the text or included amongst the exercise, all of which are supplied with solutions. The chapter on permutation groups contains a detailed account of the characters of the symmetric group based on the generating function of Frobenius and on the Schur functions. The exposition has been made self-sufficient by the inclusion of auxiliary material on skew-symmetric polynomials, determinants and symmetric functions.
Author(s): Walter Ledermann
Edition: 2
Publisher: Cambridge University Press
Year: 1987
Language: English
Pages: 239
INTRODUCTION TO GROUP CHARACTERS, 2ND ED.......Page 1
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Preface to the Second Edition......Page 8
Preface......Page 10
1.1. Introduction......Page 12
1.2. G-modules......Page 16
1.3. Characters......Page 20
1.4. Reducibility......Page 21
1.5. Permutation representations......Page 28
1.6. Complete reducibility......Page 31
1.7. Schur's Lemma......Page 35
1.8. The commutant (endomorphism) algebra......Page 38
Exercises......Page 45
2.1. Orthogonality relations......Page 48
2.2. The group algebra......Page 53
2.3. The character table......Page 60
2.4. Finite Abelian groups......Page 63
2.5. The lifting process......Page 68
2.6. Linear characters......Page 73
Exercises......Page 75
3.1. Induced representations......Page 80
3.2. The reciprocity law......Page 84
3.3. The alternating group A 5......Page 86
3.4. Normal subgroups......Page 90
3.5. Tensor products......Page 94
3.6. Mackey's Theorem......Page 105
Exercises......Page 108
4.1. Transitive groups......Page 112
4.2. The symmetric group......Page 117
4.3. Induced characters of S n......Page 119
4.4. Generalised characters......Page 121
4.5. Skew-symmetric polynomials......Page 122
4.6. Generating functions......Page 125
4.7. Orthogonality......Page 127
4.8. The degree formula......Page 131
4.9. Schur functions......Page 134
4.10. Conjugate partitions......Page 139
4.11. The characters of S 5......Page 146
Exercises......Page 149
5.1. Algebraic numbers......Page 152
5.2. Representations of the group algebra......Page 157
5.3. Burnside's (p, q)-theorem......Page 159
5.4. Frobenius groups......Page 163
Exercises......Page 169
6.1. Real character values......Page 171
6.2. Rational character values......Page 172
6.3. A congruence property......Page 176
Exercises......Page 179
7.1. Statement of the problem......Page 181
7.2. Quadratic forms......Page 182
7.3. Orthogonal representations......Page 184
7.4. Bilinear invariants......Page 188
7.5. The character criterion......Page 192
Exercises......Page 195
A.1. A generalisation of Vandermonde's determinant......Page 197
A.2. The alternant quotient......Page 199
A.3. Jacobi's Theorem on inverse matrices......Page 202
A.4. Quadratic forms......Page 205
A.5. Congruence relations in an algebraic number field......Page 208
List of character tables......Page 216
Solutions to Exercises on Chapter 1......Page 217
Solutions to Exercises on Chapter 2......Page 219
Solutions to Exercises on Chapter 3......Page 222
Solutions to Exercises on Chapter 4......Page 226
Solutions to Exercises on Chapter 5......Page 229
Solutions to Exercises on Chapter 6......Page 231
Solutions to Exercises on Chapter 7......Page 232
Bibliography......Page 235
Index......Page 237
Back Cover......Page 239
