Ⅰ—THE ORDINARY AND THE MODULAR REPRESENTATION THEORY OF A FINITE GROUP 1
Part 1:The ordinary representation theory 1
Introduction 1
11．1 Permutation representations 1
11．2 The group algebra over a field F 6
11．3 Character theory 8
11．4 Applications 17
11．5 Induced representations 19
Part 2:The modular representation theory 21
Introduction 21
12．1 The decomposition matrix D 22
12．2 Modular characters 25
12．3 Indecomposable and modularly irreducible representations 26
12．4 Character relations 29
12．5 Blocks 31
12．6 The Nakayama reciprocity formulae 33
Ⅱ—ORDINARY REPRESENTATION THEORY OF ？n AND YOUNG＇S RAISING OPERATOR 35
Introduction 35
2．1 Young＇s representation theory of ？n 35
2．2 Young＇s raising operator Rik 39
2．3 The degree fλ and the hook graph H[λ] 41
2．4 Lattice permutations 45
2．5 Skew diagrams 48
Ⅲ—？n AND THE FULL LINEAR GROUP GL(d) 52
Introduction 52
3．1 Inducing and restricting 54
3．2 The irredueible representation of GL(d) 57
3．3 The outer product [μ]．[ν] 61
3．4 The inner product [α]×[β] 64
3．5 Symmetrized outer products [μ]⊙[ν] 66
3．6 Symmetrized inner products [μ]？[ν] 71
Ⅳ—THE CHARACTERS OF ？n AND THE CONTENT OF [λ] 74
Introduction 74
4．1 Character of a cycle 75
4．2 Characters of ？n 77
4．3 The content of [λ] 80
4．4 The q-hook structure of [λ] 82
4．5 Character of a product of mq-cycles 85
4．6 The translation operator T 87
Ⅴ—THE p-BLOCK STRUCTURE OF ？n 90
Introduction 90
5．1 Construction of [λ] ftom [λ]q and the q-core[λ] 90
5．2 The hook graph H[λ] 93
5．3 The blocks of ？n 96
5．4 The primeness of q 99
Ⅵ—THE DIMENSIONS OF A p—BLOCK 101
Introduction 101
6．1 r-inducing and r-restricting 101
6．2 The r-Boolean algebra associated with [λ] 105
6．3 q-regular and q-singular Young diagrams 109
6．4 p-regular diagrams and p-regular classes of ？n 114
6．5 The number of modular irreducible representations in a block 116
Ⅵ—THE INDECOMPOSABLES OF ？n 120
Introduction 120
7．1 The D-matrix of a block of weight 1 of ？n 120
7．2 The D-matrix of a block of weight 2 of ？2p 122
7．3 Standard tableaux 124
7．4 The raising operator 126
7．5 Head and foot diagrams 130
7．6 The Nakayama reciprocity formuiae 137
Ⅷ—THE MODULAR IRREDUCIBLE REPRESENTATIONS OF ？n 141
Introduction 141
8．1 Congruence properties 142
8．2 The transforming matrix L 145
8．3 The raising operator 154
8．4 Decomposition matrices 161
8．5 r-Boolean algebras 162
Notes and References 165
Bibliography 169
Appendix(Tables) 179
Index 203