## 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.