Daniel Bump

Anne Schilling

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. Request Inspection Copy Contents: IntroductionKashiwara CrystalsCrystals of TableauxStembridge CrystalsVirtual, Fundamental, and Normal CrystalsCrystals of Tableaux IIInsertion AlgorithmsThe Plactic MonoidBicrystals and the Littlewood–Richardson RuleCrystals for Stanley Symmetric FunctionsPatterns and the Weyl Group ActionThe β∞ CrystalDemazure CrystalsThe ⋆-Involution of β∞Crystals and Tropical GeometryFurther Topics Readership: Graduate students and researchers interested in understanding from a viewpoint of combinatorics on crystal base theory.

Representations and Combinatorics

Crystal Bases

Bruno Nachtergaele

Isaiah Lankham

This is an introductory textbook designed for undergraduate mathematics majors with an emphasis on abstraction and in particular, the concept of proofs in the setting of linear algebra. Typically such a student would have taken calculus, though the only prerequisite is suitable mathematical grounding. The purpose of this book is to bridge the gap between the more conceptual and computational oriented undergraduate classes to the more abstract oriented classes. The book begins with systems of linear equations and complex numbers, then relates these to the abstract notion of linear maps on finite-dimensional vector spaces, and covers diagonalization, eigenspaces, determinants, and the Spectral Theorem. Each chapter concludes with both proof-writing and computational exercises.

Anne Schilling

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