A survey of crystal graphs
John Stembridge, Univ. of Michigan Ann Arbor
There is a rich and highly-developed combinatorial theory for for Schur
functions (Young tableaux, the Littlewood-Richardson Rule, etc), but one
can
argue that it suffers from a few too many seemingly arbitrary choices and
miracles.
On the other hand, Kashiwara's theory of crystal bases for quantum groups
comes close to subsuming this theory, and at the same time is (a) canonical
and (b) has a much greater range of applicability (namely, to the
representations of semisimple Lie groups and algebras and their quantum
analogues).
The main goal of our talk will be to explain that Kashiwara's theory can
be
developed at a purely combinatorial level, and need not rely on any of the
representation theory of quantum groups. Even in type A, this leads to a
more natural understanding of the combinatorics of Schur functions. Along
the way, we hope to mention an open problem or two.