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.
General
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