On multiplicities of maximal weights of $\hat{sl}(n)$-modules

Abstract

We determine explicitly the maximal dominant weights for the integrable highest weight $\hat{sl}(n)$-modules $V((k-1)\Lambda_0 + \Lambda_s)$, $0 \leq s \leq n-1$, $ k \geq 2$. We give a conjecture for the number of maximal dominant weights of $V(k\Lambda_0)$ and prove it in some low rank cases. We give an explicit formula in terms of lattice paths for the multiplicities of a family of maximal dominant weights of $V(k\Lambda_0)$. We conjecture that these multiplicities are equal to the number of certain pattern avoiding permutations. We prove that the conjecture holds for $k=2$ and give computational evidence for the validity of this conjecture for $k >2$.