Symmetric groups and Schur algebras

The connection between the representation theory of the symmetric group G(r) and that of the general linear group GL(n) is well established. Schur [9, 10] developed the theory for representing GL(n)(C) over C, and Green [6] showed how to refine Schur's methods in order to study representations of GL(n)(K) over K, where K is an infinite field of characteristic p. The Schur algebra is used to relate the representation theory of G(r) and of GL(n). The usual view is that the representation theory of GL(n) is richer than that of G(r). This emphasis is largely justified, but there is an argument for saying that the two groups should be treated on an equal footing, and we outline in this note how the Schur algebra fits naturally into the representation theory of the symmetric group. The approach we adopt can be amended to cover more general algebras, such as the q-Schur algebras. Our objective is to demonstrate briefly the methods which are used, without going into detailed proofs. Hence we unashamedly choose to concentrate on special cases which contain enough complexity to convey how the general arguments should go.