Combinatorics of Character Formulas for the Lie Superalgebra $\fgl(m,n).$

Abstract

Let $\fg$ be the Lie superalgebra $\fgl(m,n).$ Algorithms for computing the composition factors and multiplicities of Kac modules for $\fg$ were given by the second author in 1996, and by J. Brundan in 2003. 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 non-split 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.