Costandard modules over schur superalgebras in characteristic p

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 λ_m = 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.