MIT Combinatorics Seminar
Generalizing the Combinatorics of Young
Tableaux to Arbitrary Lie Type
Cristian Lenart (SUNY Albany)
Wednesday, April 19, 2006 4:30 PM Building 2 Room
105
(Joint with Lie Group Seminar)
ABSTRACT

Young tableaux provide a combinatorial model for the irreducible
characters of the Lie
algebra of type $A$. A simple combinatorial model for the irreducible
characters of an
arbitrary semisimple Lie algebra (and, more generally, of a symmetrizable
KacMoody
algebra) was recently developed in joint work with A. Postnikov. This
model is based
on the combinatorics of the corresponding Weyl group and, in the finite
case, of the
affine Weyl group. It leads to an extensive generalization of the
combinatorics of
Young tableaux. In this talk, we present recent developments in this
direction. We use
the setup of crystal graphs, which are colored directed graphs on the
canonical basis
of a representation. In this context, we present an explicit combinatorial
description
(generalizing Sch\"utzenberger's ``evacuation'' procedure for tableaux) of
the
involution which realizes the crystals as selfdual posets. We also
discuss
combinatorial aspects related to the product of crystals. The talk will be
largely
selfcontained.


