Multiplicities of maximal dominant weights of some affine Lie algebra modules

Abstract:

Affine Lie algebras are infinite dimensional analogs of finite dimensional simple Lie algebras. Let $s\ell (n,\mathbb{C})$ be the finite dimensional simple Lie algebra of trace zero $n\times n$ matrices. The associated affine Lie algebra $\widehat{s\ell }(n,\mathbb{C})$ is a one dimensional central extension of the loop algebra $s\ell (n,\mathbb{C})\otimes \mathbb{C}[t,t^{-1}]$. Unlike the finite dimenisonal simple Lie algebras, the irreducible highest weight modules for affine Lie algebras are infinite dimensional and have infinitely many weights. However, each weight space is finite dimensional. So it is an important problem to determine the dimensions of these weight spaces which are called the multiplicity of the corresponding weight. In this talk we will discuss the multiplicity of the maximal dominant weights of the irreducible highest weight $\widehat{s\ell }(n,\mathbb{C})$-module $V(k{\mathrm{\Lambda }}_0)$. We will use combinatorics of crystal bases, colored Young Tableaux and RSK correspondence to show that these multiplicities are equal to the number of certain pattern avoiding permutations.

Note: This talk will be held in Thomson Hall 202 instead of the usual location.

Join Zoom Meeting: https://washington.zoom.us/j/91547335974
Meeting ID: 915 4733 5974