# Crystal embeddings and fermionic formulas

## ABSTRACT

Around 1990 Kashiwara introduced crystal bases which are bases for representations of quantum algebras $U_q(g)$ as $q\to 0$ with very beautiful combinatorial properties. We will review the basic notion of crystals and discuss several embeddings of finite-dimensional affine crystals into crystals of type $A^{(1)}_n$. These embeddings are used to prove fermionic formulas for the one-dimensional configuration sums of the corresponding crystals.

This talk is based on work in collaboration with Masato Okado and Mark Shimozono.

Speaker's Contact Info: anne(at-sign)math.mit.edu