Infinite-dimensional Lie superalgebras and hook Schur functions

Details Infinite-dimensional Lie superalgebras and hook Schur functions Infinite-dimensional Lie superalgebras and hook Schur functions

2002-06-05

arxiv.org/abs/math/0206034v2

Commun. Math. Phys. 238 (2003), No. 1-2, 95-118.

Mathematics

Representation Theory

27 pages, LaTeX format

Scientific paper

10.1007/s00220-003-0819-3

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.

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