Multiplicities of some maximal dominant weights of the $\widehat{s\ell }(n)$-modules $V(k{\Lambda }_0)$

Abstract

For $n\ge 2$ consider the affine Lie algebra $\widehat{s\ell }(n)$ with simple roots $\{{\alpha }_i\mid 0\le i\le n-1\}$. Let $V(k{\Lambda }_0),k\in {\mathbb{Z}}_{\ge 1}$ 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 $k{\Lambda }_0-{\lambda }_{a,b}^{\ell }$ where ${\lambda }_{a,b}^{\ell }=\ell {\alpha }_0+(\ell -b){\alpha }_1+(\ell -(b+1)){\alpha }_2+\cdots +{\alpha }_{\ell -b}+{\alpha }_{n-\ell +a}+2{\alpha }_{n-\ell +a+1}+\dots +(\ell -a){\alpha }_{n-1}$, and $k\ge a+b$, $a,b\in {\mathbb{Z}}_{\ge 1}$, $\max \{a,b\}\le \ell \le \lfloor \frac{n+a+b}{2}\rfloor -1$. We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of $\{1,2,3,\dots \ell \}$.