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

点击翻译按钮获取中文摘要

The PI intends to undertake a combinatorial study of structuresarising from affine algebras with applications to mathematicalphysics, representation theory and q-series.The primary combinatorial objects are crystal graphs which arecolored directed graphs. They provide a combinatorialdescription of the deep theory of crystal bases of modules overquantized universal enveloping algebras developed by Kashiwaraand Lusztig: As the quantum parameter q tends to zero,these bases are described precisely by the crystal graphs encodingnearly all the essential algebraic data. Despite their importance,little is known about the combinatorial structure of crystalscorresponding 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, branchingfunctions and fusion coefficients in conformal field theory andstatistical mechanics, and the theory of symmetric functions.This is a project in combinatorial representation theory with applications to mathematical (theoretical) physics. RepresentationTheory is the area of mathematics most intimately involved withstudying the nature of symmetries, of any sort, whereever they occur.Combinatorial representation theory refers to a methodology thatuses explicitly computable formulas. About a decade ago it wasrealized that certain models in the area of physics known asstatistical mechanics have symmetries that are not yet completelyunderstood. This project carries out acombinatorial study of the structure of those classes of symmetriesthat can be represented by "colored directed graphs."Remarkably, the structures that occur have applications inmany 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 theparticle structure of the underlying physical model.