Representation Theory of The Symmetric GroupPDF电子书下载

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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

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