Multiplicities of Some Maximal Dominant Weights of the $\widehat {s\ell }(n)$-Modules V (kΛ₀)

Abstract

For n ≥ 2 consider the affine Lie algebra $\widehat {s\ell }(n)$ with simple roots {α_i∣0 ≤ i ≤ 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 kΛ₀. It is known that there are finitely many maximal dominant weights of V (kΛ₀). Using the crystal base realization of V (kΛ₀) and lattice path combinatorics we examine 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 ≥ a + b, $a,b \in \mathbb {Z}_{\geq 1}$, $\max \limits \{a,b\} \leq \ell \leq \left \lfloor \frac {n+a+b}{2} \right \rfloor -1$. We obtain two formulae to obtain these weight multiplicities - one in terms of certain standard Young tableaux and the other in terms of certain pattern-avoiding permutations.