Combinatorics of Coxeter groups

✍ Scribed by Bjorner A., Brenti F.

• Publisher: Springer
• Year: 2005
• Tongue: English
• Leaves: 378
• Series: Graduate texts in mathematics 0231
• Category: Library

✦ Synopsis

Includes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

✦ Table of Contents

Cover......Page 1
Series: Graduate Texts in Mathematics 231......Page 2
Combinatorics of Coxeter Groups......Page 4
Copyright......Page 5
Contents......Page 8
Foreword......Page 12
Notation......Page 14
1.1 Coxeter systems......Page 16
1.2 Examples......Page 19
1.3 A permutation representation......Page 26
1.4 Reduced words and the exchange property......Page 29
1.5 A characterization......Page 33
Exercises......Page 37
Notes......Page 39
2.1 Definition and first examples......Page 42
2.2 Basic properties......Page 48
2.3 The finite case......Page 51
2.4 Parabolic subgroups and quotients......Page 53
2.5 Bruhat order on quotients......Page 57
2.6 A criterion......Page 60
2.7 Interval structure......Page 63
2.8 Complement: Short intervals......Page 70
Exercises......Page 72
Notes......Page 78
3.1 Weak order......Page 80
3.2 The lattice property......Page 85
3.3 The word property......Page 90
3.4 Normal forms......Page 93
Exercises......Page 99
Notes......Page 102
4.1 A linear representation......Page 104
4.2 The geometric representation......Page 108
4.3 The numbers game......Page 112
4.4 Roots......Page 116
4.5 Roots and subgroups......Page 120
4.6 The root poset......Page 123
4.7 Small roots......Page 128
4.8 The language of reduced words is regular......Page 132
4.9 Complement: Counting reduced words and small roots......Page 136
Exercises......Page 140
Notes......Page 145
5.1 Introduction and review......Page 146
5.2 Reflection orderings......Page 151
5.3 R-polynomials......Page 155
5.4 Lattice paths......Page 164
5.5 Kazhdan-Lusztig polynomials......Page 167
5.6 Complement: Special matchings......Page 173
Exercises......Page 177
Notes......Page 185
6. Kazhdan-Lusztig representations......Page 188
6.1 Review of background material......Page 189
6.2 Kazhdan-Lusztig graphs and cells......Page 190
6.3 Left cell representations......Page 195
6.4 Knuth paths......Page 200
6.5 Kazhdan-Lusztig representations for S_n......Page 203
6.6 Left cells for S_n......Page 206
6.7 Complement: W-graphs......Page 211
Exercises......Page 213
Notes......Page 215
7.1 Poincare series......Page 216
7.2 Descents and length generating functions......Page 223
7.3 Dual equivalence and promotion......Page 229
7.4 Counting reduced decompositions in S_n......Page 237
7.5 Stanley symmetric functions......Page 247
Exercises......Page 249
Notes......Page 257
8.1 Type B......Page 260
8.2 Type D......Page 267
8.3 Type \tilde{A}......Page 275
8.4 Type \tilde{C}......Page 282
8.5 Type \tilde{B}......Page 290
8.6 Type \tilde{D}......Page 296
Exercises......Page 301
Notes......Page 308
A1: Classification of finite and affine Coxeter groups......Page 310
A2.1 Graphs and Digraphs......Page 314
A2.2 Posets......Page 315
A2.3 Simplicial complexes......Page 317
A2.4 Shellability......Page 318
A2.5 Regular CW complexes......Page 320
A3.1 Permutations......Page 322
A3.2 Tableaux......Page 324
A3.3 The Robinson-Schensted correspondence......Page 326
A3.5 Special tableaux......Page 327
A3.6 Knuth equivalence......Page 328
A3.7 Jeu de taquin slides......Page 329
A3.8 Evacuation and antievacuation......Page 330
A3.9 Symmetries of the R-S correspondence......Page 331
A3.10 Dual equivalence......Page 332
A4.1 Formal power series......Page 334
A4.2 Symmetric functions......Page 336
Bibliography......Page 338
Index of notation......Page 368
Index......Page 374