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