Representation Theory and Combinatorics

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The proposed research problems all lie in the field of combinatorial representation theory and have, as primary goals, combinatorial understanding of representations of certain rigid algebraic structures, all of which come from, or are generalizations of, Lie theoretic objects. The goals of the proposed research projects are to (1) construct a q-analogue of partition algebras, (2) generalize the Springer correspondence to finite complex reflection groups, (3) provide detailed analysis of generalized Schur functors at roots of unity, and (4) find a purely positive formula for q-weight multiplicities. This research strongly uses both algebraic geometric and combinatorial tools. Most symmetry groups are complex enough that it is difficult to study them directly and a representation is a way of extracting information about these symmetries. A general representation is a large and complex structure like a molecule and is composed of smaller ``irreducible components'' which are analogous to atoms. The goal of Combinatorial Representation Theory (and this proposal) is to find elementary models which allow us to (more) easily determine properties of these representations. These models enable one to obtain very explicit information about the fine structure of the corresponding representations, which have complexities on the order of the microscopic structure of living cells.