Combinatorics of character formulas for the lie superalgebra $ \mathfrak{g}\mathfrak{l}\left( {m,n} \right) $

Abstract

Let \( \mathfrak{g} \) be the Lie superalgebra \( \mathfrak{g}\mathfrak{l}\left( {m,n} \right) \). Algorithms for computing the composition factors and multiplicities of Kac modules for \( \mathfrak{g} \) were given by the second author, [12] and by J. Brundan [1]. We give a combinatorial proof of the equivalence between the two algorithms. The proof uses weight and cap diagrams introduced by Brundan and C. Stroppel, and cancelations between paths in a graph \( \mathcal{G} \) defined using these diagrams. Each vertex of \( \mathcal{G} \) corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If \( \mathcal{E} \) is the subgraph of \( \mathcal{G} \) obtained by deleting all edges of positive weight, then \( \mathcal{E} \) is the graph that describes nonsplit extensions between simple highest weight modules. We also give a procedure for finding the composition factors of any Kac module, without cancelation. This procedure leads to a second proof of the main result.