A combinatorial model for Kirillov-Reshetikhin crystals and applications

Cristian Lenart

Max-Planck-Institut fur Mathematik (Germany) and State University of New York at Albany (USA)

Abstract

Crystals are colored directed graphs encoding information about Lie algebra representations. Certain crystals for affine Lie algebras called Kirillov-Reshetikhin (KR) crystals are graded by the energy function. Given a tensor product of KR crystals, there is a unique crystal isomorphism commuting tensor factors, which is called the combinatorial R-matrix. In Lie type A, the vertices of a tensor product of KR crystals are indexed by column-strict fillings (with 1,...,n) of a Young diagram. The energy is computed by the Lascoux-Schutzenberger charge statistic on fillings. The combinatorial R-matrix is realized by Schutzenberger's sliding game (jeu de taquin) on two columns. I will present a combinatorial model which generalizes these constructions uniformly across all Lie types. The talk is largely self-