On Multiplicities of Maximal Weights of $\widehat {sl}(n)$-Modules

Abstract

We determine explicitly the maximal dominant weights for the integrable highest weight $\widehat {sl}(n)$-modules V((k − 1)Λ ₀ + Λ _s ), 0 ≤ s ≤ n − 1, k ≥ 2. We give a conjecture for the number of maximal dominant weights of V(k Λ 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 Λ 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.