Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(kΛ_0)$ (2004.14470v1)

Abstract: For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots ${\alpha_i \mid 0 \leq i \leq n-1}$. Let $V(k\Lambda_0), \, k \in \mathbb{Z}{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module with highest weight $k\Lambda_0$. It is known that there are finitely many maximal dominant weights of $V(k\Lambda_0)$. Using the crystal base realization of $V(k\Lambda_0)$ and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form $k\Lambda_0 - \lambda^\ell{a,b}$ where $ \lambda^\ell_{a,b} = \ell\alpha_0 + (\ell-b)\alpha_1 + (\ell-(b+1))\alpha_2 + \cdots + \alpha_{\ell-b} + \alpha_{n-\ell+a} + 2\alpha_{n - \ell+a+1} + \ldots + (\ell-a)\alpha_{n-1}$, and $k \geq a+b$, $a,b \in \mathbb{Z}_{\geq 1}$, $\max{a,b} \leq \ell \leq \left \lfloor \frac{n+a+b}{2} \right \rfloor-1 $. We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of ${1, 2, 3, \ldots \ell}$.