Representation theory of Schur Algebras over rings

Abstract

In 1933 I. Schur developed a method to investigate a finite group G acting on a set D such that G contains a subgroup H acting strictly transitively on 0. His fundamental idea consisted in taking as "points" not arbitrary objects or numbers from 1 ton, but group elements. H. Wielandt has presented Schur's method in the most simple way by using the concept of a group algebra over C,. The idea is to associate to the above situation an action of G on H , such that H acts on itself by left multiplication. Then consider the stabilizer L in G of the neutral element, and let L act on H. The sums of the orbits, considered in the group ring CH, generate a submodule of CH, which in this case is actually a ring; this algebra is called a Schur algebra. Using this method, one has : G is doubly transitive if and only if the associated Schur algebra is "trivial". More generally, H. Wielandt has defined a Schur algebra as a subalgebra of the group ring CG (G finite group) associated to a suitable partition of G. A Schur algebra over (J; is semisimple, This latter problem is investigated by F. Roesler for a Schur algebra over an arbitrary field. The investigation of the characters of a Schur algebra was set up by 0. Tamaschke and F. Roesler, see [Ti] and [R]. In [T2], the author started to study Schur algebras over C in a categorical context. Our main objective is to study indecomposable modules and trace functions for Schur algebras over a commutative ring R. Roughly speaking, a Schur algebra is a subalgebra of a group ring RG associated to a suitable partit ion of G ( G a finite group). ...