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)


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 Kac-Moody 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 self-dual posets. We also discuss combinatorial aspects related to the product of crystals. The talk will be largely self-contained.