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

Let mathfrakg mathfrak{g} be the Lie superalgebra mathfrakgmathfrakl( m,n ) mathfrak{g}mathfrak{l}left( {m,n} right) . Algorithms for computing the composition factors and multiplicities of Kac modules for mathfrakg 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 G mathcal{G} defined using these diagrams. Each vertex of G mathcal{G} corresponds to a highest weight of a finite dimensional simple module, and each edge is weighted by a nonnegative integer. If E mathcal{E} is the subgraph of G mathcal{G} obtained by deleting all edges of positive weight, then E 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.