Abstract

If R is a commutative ring, G is a nite group, and H is a subgroup of G, then the centralizer algebra RGH is the set of all elements of RG that commute with all elements of H. The algebra RGH is a Hecke algebra in the sense that it is isomorphic to EndRHG(RG) = EndRHG(1H HG). The authors have been studying the representation theory of these algebras in several recent and not so recent papers [4], [5], [6], [7], [10], [11], mainly in cases where G is p-solvable and H is normal, or when G = Sn and H = Sm for n 􀀀 3 m n. Part of the original motivation was to see whether there might be a \weight conjecture" for these algebras, one that would simultaneously generalize Alperin's weight conjecture and Brauer's First Main Theorem on Blocks. This idea is explained in more detail in in [4], [5], and [6]. Also, when H is a p-subgroup these algebras play an important role in Green's approach to modular representation theory and in Puig's theory of points. Along the way, several fairly basic and general questions have come up. This paper mainly consists of counterexamples to conjectures that one might be led to make based on the evidence in our earlier papers.