《Representation Theory of The Symmetric Group》求取 ⇩

Ⅰ—THE ORDINARY AND THE MODULAR REPRESENTATION THEORY OF A FINITE GROUP1

Part 1:The ordinary representation theory1

Introduction1

11．1Permutation representations1

11．2 The group algebra over a field F6

11．3 Character theory8

11．4 Applications17

11．5 Induced representations19

Part 2:The modular representation theory21

Introduction21

12．1The decomposition matrix D22

12．2 Modular characters25

12．3 Indecomposable and modularly irreducible representations26

12．4 Character relations29

12．5 Blocks31

12．6 The Nakayama reciprocity formulae33

Ⅱ—ORDINARY REPRESENTATION THEORY OF ？n AND YOUNG＇S RAISING OPERATOR35

Introduction35

2．1Young＇s representation theory of ？n35

2．2 Young＇s raising operator Rik39

2．3 The degree fλ and the hook graph H[λ]41

2．4 Lattice permutations45

2．5 Skew diagrams48

Ⅲ—？n AND THE FULL LINEAR GROUP GL(d)52

Introduction52

3．1Inducing and restricting54

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

Introduction74

4．1Character of a cycle75

4．2 Characters of ？n77

4．3 The content of [λ]80

4．4 The q-hook structure of [λ]82

4．5 Character of a product of mq-cycles85

4．6 The translation operator T87

Ⅴ—THE p-BLOCK STRUCTURE OF ？n90

Introduction90

5．1Construction of [λ] ftom [λ]q and the q-core[λ]90

5．2 The hook graph H[λ]93

5．3 The blocks of ？n96

5．4 The primeness of q99

Ⅵ—THE DIMENSIONS OF A p—BLOCK101

Introduction101

6．1r-inducing and r-restricting101

6．2 The r-Boolean algebra associated with [λ]105

6．3 q-regular and q-singular Young diagrams109

6．4 p-regular diagrams and p-regular classes of ？n114

6．5 The number of modular irreducible representations in a block116

Ⅵ—THE INDECOMPOSABLES OF ？n120

Introduction120

7．1The D-matrix of a block of weight 1 of ？n120

7．2 The D-matrix of a block of weight 2 of ？2p122

7．3 Standard tableaux124

7．4 The raising operator126

7．5 Head and foot diagrams130

7．6 The Nakayama reciprocity formuiae137

Ⅷ—THE MODULAR IRREDUCIBLE REPRESENTATIONS OF ？n141

Introduction141

8．1Congruence properties142

8．2 The transforming matrix L145

8．3 The raising operator154

8．4 Decomposition matrices161

8．5 r-Boolean algebras162

Notes and References165

Bibliography169

Appendix(Tables)179

Index203