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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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  • 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
  • Pages:i–xii

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Chapter 1: Introduction
  • Pages:1–6

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Chapter 2: Kashiwara Crystals
  • Pages:7–30

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Chapter 3: Crystals of Tableaux
  • Pages:31–37

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Chapter 4: Stembridge Crystals
  • Pages:38–54

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

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

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Chapter 7: Insertion Algorithms
  • Pages:96–111

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

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

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Chapter 10: Crystals for Stanley Symmetric Functions
  • Pages:133–142

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

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

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Chapter 13: Demazure Crystals
  • Pages:172–177

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

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Chapter 15: Crystals and Tropical Geometry
  • Pages:195–227

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Chapter 16: Further Topics
  • Pages:228–238

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  • Pages:239–262

Free Access
  • Pages:263–279

"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."

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Sample Chapter(s)
Introduction (188 KB)