Representations of Finite Groups and Algebraic Lie Theory

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The investigators will continue their study of the representation theoryof finite groups, especially the symmetric group and its double covers, by exploiting various connections to Lie theory. These connections arise atmany 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 affineKac-Moody algebras via vertex operators, Broue's conjecture for thesymmetric group, and to further exploit the relationship between branchingrules 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 symmetriesof something. Representation theory is the study of the ways suchsymmetries 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 hasbeen some major progress in our understanding of representation theory, thanksin part to a new influx of ideas from mathematical physics. This project isconcerned in part with the representation theory of the most important of all thefinite groups, the symmetric group, which is closely related to the theory of symmetric functions. The work is expected to have applications to other areas ofmathematics, as well as a wider impact in mathematical physics, statistical mechanics and coding theory.