On Schur algebras and related algebras VI: Some remarks on rational and classical Schur algebras

In a recent paper, Dipper and Doty, [4], introduced certain finite dimensional algebras, associated with the natural module of the general linear group and its dual, which they call rational Schur algebras. We give a proof, via tilting modules, that these algebras are in fact generalised Schur algebras. Using the same technique we show that certain finite dimensional algebras associated with classical groups, introduced by Doty, [20], are quasi-hereditary algebras. A generalised Schur algebras may be viewed as a quotient of the algebra of distributions of a reductive group by a certain ideal. We give generators for this ideal.