Schur algebras for general linear and other reductive groups

Stephen Donkin

Let k be an infinite field. For positive integers n,r one has the Schur algebra S(n,r) over k. This algebra is finite dimensional and the category of S(n,r) modules is naturally equivalent to the category of modules for the general linear group GL_n(k) that are polynomial of degree r. The algebra S(n,r) is also very closely related to the group algebra of the symmetric group S_r over k and forms a bridge between the representation theory of general linear groups and the representation theory of symmetric groups over k. This is the point of view of Schur's thesis of 1901, working over a characteristic 0 field, and of Green's monograph, Polynomial Representations of GL_n, of 1980 (Second Edition with Erdmann and Shocker 2007), working over a field of arbitrary characteristic.

The Schur algebra construction may be generalized to produce a finite dimensional algebra defined by a suitable finite set of weights for an arbitrary reductive group (over an arbitrary field). In these lectures we explore the Schur algebras S(n,r) and their generalizations. In particular we describe as generalized Schur algebras the so called rational Schur algebras recently introduced by Dipper and Doty and also the application of symplectic Schur algebras to the theory of Brauer algebras by R. Tange and the speaker.