Crystal Bases

Representations and Combinatorics

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.

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.
Key Features:

• First textbook that approaches crystal base theory solely from the combinatorial perspective
• The presentation uses the Stembridge local axioms and virtual crystals to uniquely characterize classical crystals
• The textbook incorporates examples on how to compute and experiment with crystals using the open-source software system Sage