Crystal Bases: Representations And Combinatorics

by Daniel Bump, Anne Schilling

Overview

This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.

Product Details

ISBN-13:	9789814733441
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	03/08/2017
Pages:	292
Product dimensions:	6.60(w) x 9.50(h) x 0.60(d)

Table of Contents

Preface v

Acknowledgments vii

1 Introduction 1

2 Kashiwara Crystals 7

2.1 Root systems 7

2.2 Kashiwara crystals 12

2.3 Tensor products of crystals 18

2.4 The signature rule 22

2.5 Root strings 23

2.6 The character 24

2.7 Related crystals and twisting 25

2.8 Dynkin diagrams and Levi branching 26

Exercises 28

3 Crystals of Tableaux 31

3.1 Type A crystals of tableaux 31

3.2 An example 35

Exercises 36

4 Stembridge Crystals 38

4.1 Motivation and examples 38

4.2 Stembridge axioms 42

4.3 Stembridge crystals as a monoidal category 45

4.1 Properties of Stembridge crystals 51

Exercises 54

5 Virtual, Fundamental, and Normal Crystals 55

5.1 Embeddings of root systems 55

5.2 Virtual crystals 57

5.3 Properties of virtual crystals 61

5.4 Fundamental crystals 63

5.5 Adjoint crystals 66

5.6 Fundamental crystals: The exceptional cases 68

5.7 Normal crystals 71

5.8 Reducible Cartan types 74

5.9 Similarity of crystals 75

5.10 Levi branching of normal crystals 76

Exercises 78

6 Crystals of Tableaux II 80

6.1 Column reading in type A 82

6.2 Crystals of columns 83

6.3 Crystals of tableaux 88

6.3.1 Crystal of tableaux: Type C_r 88

6.3.2 Crystal of tableaux: Type B_r 91

6.3.3 Crystal of tableaux: Type D_r 93

Exercises 95

7 Insertion Algorithms 96

7.1 The RSK algorithm 96

7.2 The dual RSK algorithm 105

7.3 Edelman-Greene insertion 107

Exercises 110

8 The Plactic Monoid 112

5.1 The definition of the plactic monoid 112

8.1 The plactic monoid and Knuth equivalence 113

8.2 Crystals and Schensted insertion 116

8.3 Crystals of skew tableaux 120

Exercises 123

9 Bicrystals and the Littlewood Richardson Rule 125

9.1 The GL(n) x GL(r) bicrystal 127

9.2 The crystal see-saw and the Littlewood-Richardson rule 130

Exercise 132

10 Crystals for Stanley Symmetric Functions 133

10.1 Stanley symmetric functions 133

10.2 Crystal on decreasing factorizations 135

10.3 Applications 137

Exercises 142

11 Patterns and the Weyl Group Action 143

11.1 String patterns 144

11.2 Gelfand-Tsetlin patterns 149

11.3 The Weyl group action 151

Exercises 156

12 The B∞ Crystal 157

12.1 Elementary crystals 158

12.2 The crystal B∞ for simply-laced types 159

12.3 The crystal B∞ for nom-simply-laced types 164

12.4 Demazure crystals in B∞ 167

Exercises 170

13 Demazure Crystals 172

13.1 Demazure operators and the Demazure character formula 172

13.2 Demazure crystals 174

13.3 Crystal Demazure operators 174

Exercises 176

14 The *-Involution of B∞ 178

14.1 The A₂ case 179

14.2 The general case 180

14.3 Properties of the involution 190

14.3.1 Relation to Demazure crystals 190

14.3.2 Characterization of highest weight crystals 191

14.3.3 Committor 192

Exercises 193

15 Crystals and Tropical Geometry 195

15.1 Lusztig parametrization: The A₂ case 197

15.2 Geometiic preparations 199

15.3 The Lusztig parametrization in the simply-laced case 201

15.4 Weyl group action 205

15.5 The geometric weight map 207

15.6 MV polytopes: The A₂ ease 212

15.7 Tropical Plücker relations 213

15.8 The crystal structure on MV polytopes 217

15.9 The *-involution 221

15.10 MV polytopes and the finite crystals Bλ 223

Exercises 224

16 Further Topics 228

16.1 Kirillov-Reshetikhin crystals 228

16.2 Littelmann path and alcove path models 230

16.3 Kyoto path model 231

16.4 Nakajima monomial model 232

16.5 Crystals on rigged configurations 233

16.6 Modular branching rules of the symmetric group and crystal bases 234

16.7 Tokuyama's formula 234

16.8 Crystals of Lie superalgebras 236

Appendix A Schur-Weyl Duality 239

A.1 Generalities 239

A.2 The Schur-Weyl duality correspondence 246

A.3 Symmetric functions 248

A.4 See-saws 250

Appendix B The Cauchy Correspondence 252

B.1 The Cauchy identity 252

B.2 Three interpretations of Littlewood-Richardson coefficients 254

B.3 Pieri's formula 256

B.4 Symmetric group branching rules 258

B.5 The involution on symmetric functions 260

B.6 The GL(n,C) branching rule 260

B.7 The dual Cauchy identity 262

Bibliography 263

Index 275