On Multiplicities of Maximal Weights of sl̂(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widehat {sl}(n)$\end{document}-Modules

We determine explicitly the maximal dominant weights for the integrable highest weight sl̂(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widehat {sl}(n)$\end{document}-modules V((k − 1)Λ0 + Λs), 0 ≤ s ≤ n − 1, k ≥ 2. We give a conjecture for the number of maximal dominant weights of V(kΛ0) 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(kΛ0). 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.