COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

Abstract

In this paper we consider the problem of describing the costandard modules ∇(λ) of a Schur superalgebra S(m|n,r) over a base field K of arbitrary characteristic. Precisely, if G = GL(m|n) is a general linear supergroup and Dist (G) its distribution superalgebra we compute the images of the Kostant ℤ-form under the epimorphism Dist (G) → S(m|n,r). Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain an isomorphism ∇(λ) ≈ ∇(λ+|0) ⊗ ∇(0|λ-) assuming that λ = (λ+|λ-) and λm = 0. If char (K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ)p where ∇(μ) is a costandard module of the ordinary Schur algebra S(n,r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux.