Representations and Cohomology of Groups

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

Absract for Benson's NSF Proposal 9988110 Professor Benson's research centers around group representation theory and its connections with cohomology. In recent years, infinite dimensional representations of finite groups have played an increasingly important role, especially because of the idempotent modules of Rickard. Benson proposes to work on the role of the nucleus for nonprincipal blocks, pure injectivity and phantom maps, and their connections with Grothendieck's local duality, and infinitely generated endotrivial modules in this context. He also proposes to work with Greenlees on duality spectral sequences for p-adic analytic groups with torsion, and with Carlson on calculating cohomology of some specific finite groups, as well as continuing his investigation of 2-completions of classifying spaces as homotopy colimits. Representation theory, in the context of this proposal, means the representations of abstract groups as groups of matrices. Cohomology measures how these matrices can be patched together, as well as other more subtle properties of the matrix representations. Progress in this area has implications in algebra, topology, geometry and number theory. Recent developments using infinite dimensional representations have had a great impact on our understanding of the finite dimensional ones. Benson's research involves theoretical and computational aspects of finite and infinite dimensional representations and their cohomology, making use of recent progress to push forward our understanding of these and related areas.