Infinite-dimensional Lie superalgebras and hook Schur functions

Abstract

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

27 pages, LaTeX format

Cited by in corpus (6)

• Howe duality and Kostant Homology Formula for infinite-dimensional Lie superalgebras
• The Bloch-Okounkov correlation functions of classical type
• Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras
• An integrable $U_q(\widehat{gl}(2|2){)}_1$-Model: Corner Transfer Matrices and Young Skew Diagrams
• Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras
• Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras