Affine Lie algebra weight multiplicities and pattern avoiding permutations

Affine Lie algebras form an important class of Kac-Moody Lie algebras. The combinatorial representation theory of affine Lie algebras has many applications in other areas of mathematics and physics. It is known that there are finitely many maximal dominant weights of an irreducible representation of an affine Lie algebra. However, it is a nontrivial problem to determine their multiplicities. In this talk we will discuss the multiplicities of maximal dominant weights of the highest weight $sl^(n)$- module $V(k\Lambda_0)$.

We use the extended Young diagram realizations of the crystal bases of these modules to show that these multiplicities are given by certain pattern avoiding permutations. This talk is based on some joint work with Rebecca Jayne.

Bio
https://misra.math.ncsu.edu/