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El. knyga: Crystal Bases: Representations And Combinatorics

(Stanford Univ, Usa), (Univ Of California, Davis, Usa)
  • Formatas: 292 pages
  • Išleidimo metai: 17-Jan-2017
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789814733465
Kitos knygos pagal šią temą:
Crystal Bases: Representations And CombinatoricsDidesnis paveikslėlis
  • Formatas: 292 pages
  • Išleidimo metai: 17-Jan-2017
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789814733465
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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 the ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear group, and phenomena in combinatorics. The authors are both contributors to Sage, an open-source mathematical software system, which has strong support for crystal bases and combinatorics and the book takes advantage of this.
Preface v
Acknowledgments vii
1 Introduction
1(6)
2 Kashiwara Crystals
7(24)
2.1 Root systems
7(5)
2.2 Kashiwara crystals
12(6)
2.3 Tensor products of crystals
18(4)
2.4 The signature rule
22(1)
2.5 Root strings
23(1)
2.6 The character
24(1)
2.7 Related crystals and twisting
25(1)
2.8 Dynkin diagrams and Levi branching
26(5)
Exercises
28(3)
3 Crystals of Tableaux
31(7)
3.1 Type A crystals of tableaux
31(4)
3.2 An example
35(3)
Exercises
36(2)
4 Stembridge Crystals
38(17)
4.1 Motivation and examples
38(4)
4.2 Stembridge axioms
42(3)
4.3 Stembridge crystals as a monoidal category
45(6)
4.4 Properties of Stembridge crystals
51(4)
Exercises
54(1)
5 Virtual, Fundamental, and Normal Crystals
55(25)
5.1 Embeddings of root systems
55(2)
5.2 Virtual crystals
57(4)
5.3 Properties of virtual crystals
61(2)
5.4 Fundamental crystals
63(3)
5.5 Adjoint crystals
66(2)
5.6 Fundamental crystals: The exceptional cases
68(3)
5.7 Normal crystals
71(3)
5.8 Reducible Cartan types
74(1)
5.9 Similarity of crystals
75(1)
5.10 Levi branching of normal crystals
76(4)
Exercises
78(2)
6 Crystals of Tableaux II
80(16)
6.1 Column reading in type A
82(1)
6.2 Crystals of columns
83(5)
6.3 Crystals of tableaux
88(8)
6.3.1 Crystal of tableaux: Type Cr
88(3)
6.3.2 Crystal of tableaux: Type Br
91(2)
6.3.3 Crystal of tableaux: Type Dr
93(2)
Exercises
95(1)
7 Insertion Algorithms
96(16)
7.1 The RSK algorithm
96(9)
7.2 The dual RSK algorithm
105(2)
7.3 Edelman--Greene insertion
107(5)
Exercises
110(2)
8 The Plactic Monoid
112(13)
8.1 The definition of the plactic monoid
112(1)
8.2 The plactic monoid and Knuth equivalence
113(3)
8.3 Crystals and Schensted insertion
116(4)
8.4 Crystals of skew tableaux
120(5)
Exercises
123(2)
9 Bicrystals and the Littlewood--Richardson Rule
125(8)
9.1 The GL(n) × GL(r) bicrystal
127(3)
9.2 The crystal see-saw and the Littlewood--Richardson rule
130(3)
Exercise
132(1)
10 Crystals for Stanley Symmetric Functions
133(10)
10.1 Stanley symmetric functions
133(2)
10.2 Crystal on decreasing factorizations
135(2)
10.3 Applications
137(6)
Exercises
142(1)
11 Patterns and the Weyl Group Action
143(14)
11.1 String patterns
144(5)
11.2 Gelfand--Tsetlin patterns
149(2)
11.3 The Weyl group action
151(6)
Exercises
156(1)
12 The B∞ Crystal
157(15)
12.1 Elementary crystals
158(1)
12.2 The crystal B∞ for simply-laced types
159(5)
12.3 The crystal B∞ for non-simply-laced types
164(3)
12.4 Demazure crystals in B∞
167(5)
Exercises
170(2)
13 Demazure Crystals
172(6)
13.1 Demazure operators and the Demazure character formula
172(2)
13.2 Demazure crystals
174(1)
13.3 Crystal Demazure operators
174(4)
Exercises
176(2)
14 The *-Involution of B∞
178(17)
14.1 The A2 case
179(1)
14.2 The general case
180(10)
14.3 Properties of the involution
190(5)
14.3.1 Relation to Demazure crystals
190(1)
14.3.2 Characterization of highest weight crystals
191(1)
14.3.3 Commutor
192(1)
Exercises
193(2)
15 Crystals and Tropical Geometry
195(33)
15.1 Lusztig parametrization: The A2 case
197(2)
15.2 Geometric preparations
199(2)
15.3 The Lusztig parametrization in the simply-laced case
201(4)
15.4 Weyl group action
205(2)
15.5 The geometric weight map
207(5)
15.6 MV polytopes: The A2 case
212(1)
15.7 Tropical Plucker relations
213(4)
15.8 The crystal structure on MV polytopes
217(4)
15.9 The *-involution
221(2)
15.10 MV polytopes and the finite crystals Bλ
223(5)
Exercises
224(4)
16 Further Topics
228(11)
16.1 Kirillov--Reshetikhin crystals
228(2)
16.2 Littelmann path and alcove path models
230(1)
16.3 Kyoto path model
231(1)
16.4 Nakajima monomial model
232(1)
16.5 Crystals on rigged configurations
233(1)
16.6 Modular branching rules of the symmetric group and crystal bases
234(1)
16.7 Tokuyama's formula
234(2)
16.8 Crystals of Lie superalgebras
236(3)
Appendix A Schur--Weyl Duality
239(13)
A.1 Generalities
239(7)
A.2 The Schur--Weyl duality correspondence
246(2)
A.3 Symmetric functions
248(2)
A.4 See-saws
250(2)
Appendix B The Cauchy Correspondence
252(11)
B.1 The Cauchy identity
252(2)
B.2 Three interpretations of Littlewood--Richardson coefficients
254(2)
B.3 Pieri's formula
256(2)
B.4 Symmetric group branching rules
258(2)
B.5 The involution on symmetric functions
260(1)
B.6 The GL(n, C) branching rule
260(2)
B.7 The dual Cauchy identity
262(1)
Bibliography 263(12)
Index 275
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