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.