Howe Duality for Lie Superalgebras

Details Howe Duality for Lie Superalgebras Howe Duality for Lie Superalgebras

2000-08-13

arxiv.org/abs/math/0008093v1

Compositio Math.128 (2001) 55-94

Mathematics

Representation Theory

41 pages, LaTeX format, to appear in Compositio Math

Scientific paper

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 find as well 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.

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