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.