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COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

ROBERTO LA SCALA

Dipartimento di Matematica, via Orabona 4, 70125 Bari, Italia

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ALEXANDER ZUBKOV

Omsk State Pedagogical University, Tuhachenvskogo embankment 14, 644099 Omsk, Russia

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https://doi.org/10.1142/S0219498808002734Cited by:12 (Source: Crossref)

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 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.

Keywords:

AMSC: 17A70, 20G05, 05E15

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