On a q-analogue of the fusion product

Mark Shimozono

Virginia Tech

March 10,
refreshments at 3:45pm


For a simple Lie algebra g there is a ring homomorphism from the representation ring of its finite dimensional representations, to the ring of "level-restricted" representations that takes the usual tensor product to the fusion product. We are interested in a $q$-analogue of the tensor product that is compatible with the level restriction map; here by $q$-analogue we mean a grading or filtration. Let g be of classical type. Affine crystal theory yields a natural grading of both the ordinary and fusion tensor product multiplicity, but at the cost of replacing the irreducible tensor factors with suitable representations (which outside of type A, no longer be irreducible).

In type A, Foda, Leclerc, Okado, and Thibon [FLOT] have defined a $q$-analogue of the level restriction map, a Z[q]-linear map (no longer a ring homomorphism) that specializes to level restriction at $q=1$. We show that this map sends the q-analogue of ordinary tensor product to the q-analogue of the fusion product. Almost all of our proof works for classical type, where the FLOT map has a natural generalization. The only gap is a conjectural lemma on the grading, which is known to hold in type A.

Finally, in type A we prove a q-analogue of the level-rank duality for fusion product multiplicities in the special case where affine crystal theory applies, and propose its generalization for the q-analogue of tensor product multiplicity defined by Lascoux, Leclerc, and Thibon via ribbon tableaux.

This is joint work with Anne Schilling.

Speaker's Contact Info: mshimo(at-sign)math.vt.edu

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