Howe Duality for Lie Superalgebras

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.