On Schur algebras and related algebras, II

Abstract

This paper is a continuation of [7]. The purpose of [7] was to set up, in the context of reductive algebraic groups, a theory of “generalized Schur algebras.” A special case of our construction gives the Schur algebras S(n, r) studied extensively in [lo]. We concentrate here on the Schur algebras themselves and show that our construction yields new results in their representation theory. By means of the Schur functor (see [ 10, Chap. 6]), we obtain new results in the representation theory of the symmetric groups. The main results are as follows. The Schur algebras S,(n, r), defined over a principal ideal domain R, have finite global dimension (see also [ 1 I). Over a field or complete valuation ring R, every finitely generated projective S,(n, r)-module has a Weyl module filtration. We prove the irreducibility of the socle of certain Weyl modules (see also [15]). We describe a connection between weights of projective indecomposable modules for the Schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation. We show that, over a complete valuation ring or a field, every Young module for a symmetric group has a Specht series, in particular the projective indecomposables have Specht series. Finally, we show that, for r <n and a complete valuation ring or field R, the blocks of the Schur algebra S,(n, r) are given by the rule embodied in Nakayama’s conjecture. The notations and conventions are as in the first part of this work [7].