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

PI打算对仿射代数产生的结构进行组合研究，并将其应用于数学物理、表示理论和q级数。主要的组合对象是晶体图，它是彩色有向图。他们对kashiwara和Lusztig提出的模超量子化包络代数的晶体基的深层理论进行了组合描述：当量子参数q趋于零时，这些基被编码几乎所有基本代数数据的晶体图精确地描述。尽管它们很重要，但人们对与仿射代数有限维模块相对应的晶体组合结构知之甚少。PI提出了一种进行这种组合研究的方法。这些研究将在共形场论和统计力学中的q级数、分支函数和融合系数以及对称函数理论中得到应用。这是一个关于组合表示理论及其在数学（理论）物理中的应用的项目。表示理论是数学中与研究任何形式的对称的本质最密切相关的领域，无论它们发生在哪里。组合表示理论指的是一种使用显式可计算公式的方法。大约十年前，人们意识到，在被称为统计力学的物理领域中，某些模型具有尚未完全理解的对称性。这个项目对那些可以用“彩色有向图”表示的对称类的结构进行了组合研究。值得注意的是，这种结构在数学和物理的许多不同领域都有应用，比如统计力学模型、表示理论、对称函数理论和组合学。例如，它们导致了对底层物理模型的粒子结构进行编码的公式。

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.