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 $V((k-1){\Lambda }_0+{\Lambda }_s)$, $0\le s\le n-1$, $k\ge 2$. 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 $k>2$.

arXiv comment17 pages

References in corpus (1)

• A non-recursive criterion for weights of a highest weight module for an affine Lie algebra