The representation theory of finite groups has seen rapid growth in recent years with the development of efficient algorithms and computer algebra systems. This is the first book to provide an introduction to the ordinary and modular representation theory of finite groups with special emphasis on the computational aspects of the subject. Evolving from courses taught at Aachen University, this well-paced text is ideal for graduate-level study. The authors provide over 200 exercises, both theoretical and computational, and include worked examples using the computer algebra system GAP. These make the abstract theory tangible and engage students in real hands-on work. GAP is freely available from www.gap-system.org and readers can download source code and solutions to selected exercises from the book's web page.
Author(s): Klaus Lux, Herbert Pahlings
Series: Cambridge Studies in Advanced Mathematics 124
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 472
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 9
Frequently used symbols......Page 12
1.1 Basic concepts......Page 13
1.2 Permutation representations and G-sets......Page 35
1.3 Simple modules, the “Meataxe”......Page 48
1.4 Structure of algebras......Page 62
1.5 Semisimple rings and modules......Page 67
1.6 Direct sums and idempotents......Page 75
1.7 Blocks......Page 91
1.8 Changing coefficients......Page 95
2.1 Characters and block idempotents......Page 99
2.2 Character values......Page 113
2.3 Character degrees......Page 121
2.4 The Dixon–Schneider algorithm......Page 125
2.5 Application – generation of groups......Page 133
2.6 Character tables......Page 147
2.7 Products of characters......Page 151
2.8 Generalized characters and lattices......Page 161
2.9 Invariant bilinear forms and the Schur index......Page 176
2.10 Computing character tables – an example......Page 185
3.1 Restriction and fusion......Page 191
3.2 Induced modules and characters......Page 194
3.3 Symmetric groups......Page 211
3.4 Permutation characters......Page 218
3.5 Tables of marks......Page 222
3.6 Clifford theory......Page 235
3.7 Projective representations......Page 250
3.8 Clifford matrices......Page 271
3.9 M-groups......Page 289
3.10 Brauer’s induction theorem......Page 293
4.1 p-modular systems......Page 301
4.2 Brauer characters......Page 313
4.3 p-projective characters......Page 327
4.4 Characters in blocks......Page 333
4.5 Basic sets......Page 343
4.6 Defect groups......Page 356
4.7 Brauer correspondence......Page 362
4.8 Vertices......Page 377
4.9 Green correspondence......Page 390
4.10 Trivial source modules......Page 398
4.11 Generalized decomposition numbers......Page 408
4.12 Brauer’s theory of blocks of defect one......Page 419
4.13 Brauer characters of p-solvable groups......Page 434
4.14 Some conjectures......Page 438
References......Page 455
Notation index......Page 467
Subject index......Page 469
