Infinite-dimensional Lie superalgebras and hook Schur functions

Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\text{hgltwo}$ and the finite-dimensional group $G{L}_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\text{hgltwo}$ in terms of hook Schur functions. We use the reduction procedure of $\text{hgltwo}$ to ${\widehat{gl}}_{n\mathrm{\mid }n}$ to derive a character formula for a certain class of level 1 highest weight irreducible representations of ${\widehat{gl}}_{n\mathrm{\mid }n}$, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra $g{l}_{n\mathrm{\mid }n}$. These modules turn out to form the complete set of integrable ${\widehat{gl}}_{n\mathrm{\mid }n}$-modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible ${\widehat{gl}}_{m\mathrm{\mid }n}$-modules may be written as a sum of products of hook Schur functions.