### Abstract

Making use of a Howe duality involving the infinite-dimensional Lie super-algebra gl_{∞|∞} and finite-dimensional group GL_{l} of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of gl_{∞|∞} in terms of hook Schur functions. We use the reduction procedure of gl_{∞|∞} to gl_{n|n} to derive a character formula for a certain class of level 1 highest weight irreducible representations of 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 gl_{n|n}-modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible gl_{m|n}-modules may be written as a sum of products of hook Schur functions.

Original language | English |
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Pages (from-to) | 95-118 |

Number of pages | 24 |

Journal | Communications in Mathematical Physics |

Volume | 238 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2003 Jan 1 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*238*(1-2), 95-118. https://doi.org/10.1007/s00220-003-0819-3

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*Communications in Mathematical Physics*, vol. 238, no. 1-2, pp. 95-118. https://doi.org/10.1007/s00220-003-0819-3

**Infinite-dimensional Lie superalgebras and hook Schur functions.** / Cheng, Shun Jen; Lam, Ngau.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Infinite-dimensional Lie superalgebras and hook Schur functions

AU - Cheng, Shun Jen

AU - Lam, Ngau

PY - 2003/1/1

Y1 - 2003/1/1

N2 - Making use of a Howe duality involving the infinite-dimensional Lie super-algebra gl∞|∞ and finite-dimensional group GLl of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of gl∞|∞ in terms of hook Schur functions. We use the reduction procedure of gl∞|∞ to gln|n to derive a character formula for a certain class of level 1 highest weight irreducible representations of gln|n, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra gln|n. These modules turn out to form the complete set of integrable gln|n-modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible glm|n-modules may be written as a sum of products of hook Schur functions.

AB - Making use of a Howe duality involving the infinite-dimensional Lie super-algebra gl∞|∞ and finite-dimensional group GLl of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of gl∞|∞ in terms of hook Schur functions. We use the reduction procedure of gl∞|∞ to gln|n to derive a character formula for a certain class of level 1 highest weight irreducible representations of gln|n, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra gln|n. These modules turn out to form the complete set of integrable gln|n-modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible glm|n-modules may be written as a sum of products of hook Schur functions.

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U2 - 10.1007/s00220-003-0819-3

DO - 10.1007/s00220-003-0819-3

M3 - Article

VL - 238

SP - 95

EP - 118

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1-2

ER -