Representations of Finite Groups and Algebraic Lie Theory

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

The investigators will continue their study of the representation theory of finite groups, especially the symmetric group and its double covers, by exploiting various connections to Lie theory. These connections arise at many different levels, combinatorial, algebraic and geometric, and involve the representation theory of algebraic groups, supergroups, quantum groups and infinite dimensional Lie algebras. The investigators intend to study in particular the Shapovalov form on highest weight modules over affine Kac-Moody algebras via vertex operators, Broue's conjecture for the symmetric group, and to further exploit the relationship between branching rules and crystal graphs in representation theory. This project is in the area of mathematics known as representation theory. The tools of mathematics provide a precise way to describe the symmetries of something. Representation theory is the study of the ways such symmetries can arise in the real world, and as such it has applications to many areas of mathematics, physics and chemistry. In the last few years, there has been some major progress in our understanding of representation theory, thanks in part to a new influx of ideas from mathematical physics. This project is concerned in part with the representation theory of the most important of all the finite groups, the symmetric group, which is closely related to the theory of symmetric functions. The work is expected to have applications to other areas of mathematics, as well as a wider impact in mathematical physics, statistical mechanics and coding theory.