The Combinatorics of Affine Algebras and their Applications to Mathematical Physics and Representation Theory

University Of California-Davis, Davis CA

Investigators

Abstract

The PI intends to undertake a combinatorial study of structures arising from affine algebras with applications to mathematical physics, representation theory and q-series. The primary combinatorial objects are crystal graphs which are colored directed graphs. They provide a combinatorial description of the deep theory of crystal bases of modules over quantized universal enveloping algebras developed by Kashiwara and Lusztig: As the quantum parameter q tends to zero, these bases are described precisely by the crystal graphs encoding nearly all the essential algebraic data. Despite their importance, little is known about the combinatorial structure of crystals corresponding to finite-dimensional modules of affine algebras. The PI proposes a method to undertake such a combinatorial study. These studies will have applications to q-series, branching functions and fusion coefficients in conformal field theory and statistical mechanics, and the theory of symmetric functions. This is a project in combinatorial representation theory with applications to mathematical (theoretical) physics. Representation Theory is the area of mathematics most intimately involved with studying the nature of symmetries, of any sort, whereever they occur. Combinatorial representation theory refers to a methodology that uses explicitly computable formulas. About a decade ago it was realized that certain models in the area of physics known as statistical mechanics have symmetries that are not yet completely understood. This project carries out a combinatorial study of the structure of those classes of symmetries that can be represented by "colored directed graphs." Remarkably, the structures that occur have applications in many diverse areas of mathematics and physics, such as statistical mechanical models, representation theory, the theory of symmetric functions and combinatorics. For example, they lead to formulas which encode the particle structure of the underlying physical model.