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Howe Duality for Lie Superalgebras

Published online by Cambridge University Press:  04 December 2007

Shun-Jen Cheng  and
Weiqiang Wang
Shun-Jen Cheng
Affiliation:
Department of Mathematics, National Taiwan University, Taipei, Taiwan. E-mail: chengsj@math.ntu.edu.tw
Weiqiang Wang
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, U.S.A. E-mail: wqwang@math.ncsu.edu
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Abstract

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We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest-weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible gl(m|n)-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of gl(m|n). In the former case, we also find explicit formulas for the highest-weight vectors. Our work unifies and generalizes the classical results in symmetric and skew-symmetric models and admits several applications.

Keywords

Lie superalgebraHowe dualityhighest-weight vectors
Type
Research Article
Information
Compositio Mathematica , Volume 128 , Issue 1 , August 2001 , pp. 55 - 94
Copyright
© 2001 Kluwer Academic Publishers