Polynomial Representations of GLn Book

Polynomial representations of GL_n

1 Introduction 1

2 Polynomial Representations of GL_n (K): The Schur algebra 11

2.1 Notation, etc 11

2.2 The categories M_K(n), M_K (n, r) 12

2.3 The Schur algebra S_K(n, r) 13

2.4 The map e: KΓ → S_K (n, r) 14

2.5 Modular theory 16

2.6 The module E^⊗r 17

2.7 Contravariant duality 19

2.8 A_K(n, r) as KT-bimodule 21

3 Weights and Characters 23

3.1 Weights 23

3.2 Weights spaces 23

3.3 Some properties of weight spaces 24

3.4 Characters 26

3.5 Irreducible modules in M_K(n, r) 28

4 The modules D_{λ, K} 33

4.1 Preamble 33

4.2 λ-tableaux 33

4.3 Bideterminants 34

4.4 Definition of D_{λ, K} 35

4.5 The basis theorem for D_{λ, K} 36

4.6 The Carter-Lusztig lemma 37

4.7 Some consequences of the basis theorem 39

4.8 James's construction of D_{λ, K} 40

5 The Carter-Luszting modules V_{λ, K} 43

5.1 Definition of V_{λ, K} 43

5.2 V_{λ, K} is Carter-Luszting's "Weyl module" 43

5.3 The Carter-Lusztig basis for V_{λ, K} 45

5.4 Some consequences of the basis theorem 47

5.5 Contravariant forms on V_{λ, K} 48

5.6 Z-forms of V_{λ, K} 50

6 Representation theory of the symmetric group 53

6.1 The functor f: M_K (n, r) → mod KG(r) (r ≤ n) 53

6.2 General theory of the functor f: mod S mod eSe 55

6.3 Application I. Specht modules and their duals 57

6.4 Application II. Irreducible KG(r)-modules, char K = p 60

6.5Application III. The functor f: M_K (N, r) → M_K (n, r) (N ≥ n) 65

6.6 Application IV. Some theorems on decomposition numbers 67

Appendix: Schensted correspondence and Littelmann paths

A Introduction 73

A.1 Preamble 73

A.2 The Robinson-Schensted algorithm 74

A.3 The operators e_c, f_c 75

A.4 What is to be done 78

B The Schensted Process 81

B.1 Notations for tableaux 81

B.2 The map Sch: I(n, r) → T(n, r) 81

B.3 Inserting a letter into a tableau 82

B.4 Examples of the Schensted process 85

B.5 Proof that (&mu, U, V) ← x₁ belongs to T(n, r) 88

B.6 The inverse Schensted process 89

B.7 The ladder 92

C Schensted and Littelmann operators 95

C.1 Preamble 95

C.2 Unwinding a tableau 96

C.3 Knuth's theorem 103

C.4 The "if" part of Knuth's theorem 107

C.5 Littelmann operators on tableaux 114

C.6 The proof of Proposition B 116

D Theorem A and some of its consequences 121

D.1 Ingredients for the proof of Theorem A 121

D.2 Proof of Theorem A 124

D.3 Properties of the operator C 127

D.4 The Littelmann algebra L(n, r) 129

D.5 The modules M_Q 131

D.6 The λ-rectangle 134

D.7 Canonical maps 135

D.8 The algebra structure of L(n, r) 137

D.9 The character of M_λ 139

D.10 The Littlewood-Richardson Rule 140

D.11 Lascoux, Leclerc and Thibon 143

E Tables 147

E.1 Schensted's decomposition of I(3,3) 147

E.2 The Littelmann graph I(3,3) 148

Index of symbols 151

References 155

Index 159