Irreducible tensor representations of general linear Lie superalgebras

Abstract

Abstract. We present a description of irreducible tensor representations of general linear Lie superalgebras in terms of generalized determinants in the symmetric and exterior superalgebras of a superspace over a field of characteristic zero.1. Introduction. Tensor representations of the general linear Lie superalgebras play a special role in the theory of representations of these superalgebras because of their relationship with the representation theory of the symmetric groups and use of combinatorial methods. The first construction of irreducible tensor representations was provided independently by Sergeev [10] and by Berele and Regev [3] using the Schur-Weyl duality in a space of tensors of a superspace over a field of characteristic 0. Implicitly, a different construction was contained earlier in a paper by Akin, Buchsbaum and Weyman [1] without mentioning Lie superalgebras. See also a more recent paper by Sergeev [11] for still another approach.In the present paper we provide an approach to these representations using generalized determinants in the symmetric superalgebra of a superspace generalizing in this way the classical constructions in terms of products of minors for general linear groups, abundant in the literature (see, e.g., a paper by de Concini, Eisenbud and Procesi [4] and the bibliography cited there). The generalized determinants we use were first considered by Doubilet and Rota in [5] and by Grosshans, Rota and Stein in [7] for applications to invariant theory.The dual version that places representations in the exterior superalgebra instead of the symmetric one is only stated for record without proofs. This generalizes earlier expositions for general linear groups by Barnabei [2] and Józefiak [9]. In an effort to make this paper relatively self-contained we provide some background material in Section 2; in Section 4C we rely on