On a q-analogue of the fusion product
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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