## On a q-analogue of the fusion multiplicities

**Mark Shimozono**.

(Verginia Tech.)

Abstract: We discuss two combinatorial formulas for a q-analogue of
the level-restricted tensor product multiplicities in type A, in the case
of several tensor factors all indexed by rectangular partitions. This q-analogue
is defined using affine crystal theory. The combinatorial objects in the
two formulas are certain Littlewood-Richardson tableaux and rigged configurations.
The rigged config formulas allow us to compute some new explicit formulas
for some affine type A branching functions. We also give a q-analogue of
level-rank duality for the above crystal level-restricted q-analogues and
a conjectural generalization for the q-analogues of Lascoux-Leclerc-Thibon.

This is joint work with Anne Schilling.