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Crystal Bases cover

BookAuthority Best Combinatorics Books of All Time

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.

Sample Chapter(s)
Chapter 1: Introduction (188 KB)

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Contents:
  • Introduction
  • Kashiwara Crystals
  • Crystals of Tableaux
  • Stembridge Crystals
  • Virtual, Fundamental, and Normal Crystals
  • Crystals of Tableaux II
  • Insertion Algorithms
  • The Plactic Monoid
  • Bicrystals and the Littlewood–Richardson Rule
  • Crystals for Stanley Symmetric Functions
  • Patterns and the Weyl Group Action
  • The β Crystal
  • Demazure Crystals
  • The ⋆-Involution of β
  • Crystals and Tropical Geometry
  • Further Topics

Readership: Graduate students and researchers interested in understanding from a viewpoint of combinatorics on crystal base theory.

Free Access
FRONT MATTER
  • Pages:i–xii

https://doi.org/10.1142/9789814733458_fmatter

Free Access
Chapter 1: Introduction
  • Pages:1–6

https://doi.org/10.1142/9789814733458_0001

No Access
Chapter 2: Kashiwara Crystals
  • Pages:7–30

https://doi.org/10.1142/9789814733458_0002

No Access
Chapter 3: Crystals of Tableaux
  • Pages:31–37

https://doi.org/10.1142/9789814733458_0003

No Access
Chapter 4: Stembridge Crystals
  • Pages:38–54

https://doi.org/10.1142/9789814733458_0004

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Chapter 5: Virtual, Fundamental, and Normal Crystals
  • Pages:55–79

https://doi.org/10.1142/9789814733458_0005

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Chapter 6: Crystals of Tableaux II
  • Pages:80–95

https://doi.org/10.1142/9789814733458_0006

No Access
Chapter 7: Insertion Algorithms
  • Pages:96–111

https://doi.org/10.1142/9789814733458_0007

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Chapter 8: The Plactic Monoid
  • Pages:112–124

https://doi.org/10.1142/9789814733458_0008

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Chapter 9: Bicrystals and the Littlewood–Richardson Rule
  • Pages:125–132

https://doi.org/10.1142/9789814733458_0009

No Access
Chapter 10: Crystals for Stanley Symmetric Functions
  • Pages:133–142

https://doi.org/10.1142/9789814733458_0010

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Chapter 11: Patterns and the Weyl Group Action
  • Pages:143–156

https://doi.org/10.1142/9789814733458_0011

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Chapter 12: The B Crystal
  • Pages:157–171

https://doi.org/10.1142/9789814733458_0012

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Chapter 13: Demazure Crystals
  • Pages:172–177

https://doi.org/10.1142/9789814733458_0013

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Chapter 14: The ★-Involution of B
  • Pages:178–194

https://doi.org/10.1142/9789814733458_0014

No Access
Chapter 15: Crystals and Tropical Geometry
  • Pages:195–227

https://doi.org/10.1142/9789814733458_0015

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Chapter 16: Further Topics
  • Pages:228–238

https://doi.org/10.1142/9789814733458_0016

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Appendices
  • Pages:239–262

https://doi.org/10.1142/9789814733458_0017

Free Access
BACK MATTER
  • Pages:263–279

https://doi.org/10.1142/9789814733458_bmatter

"The introduction to crystal bases given in this book is accessible to graduate students and researchers. Its combinatorial approach is reader-friendly and invites the reader to get involved, either working on the exercises listed at the end of each chapter or by playing with the computational implementation on freely-available software, Sage in this case, where the authors have many built-in accessible examples and some algorithms for specific crystal bases."

MAA Reviews

Sample Chapter(s)
Introduction (188 KB)