Representations of Finite Groups and Algebraic Lie Theory

研究人员将继续研究有限群的表示理论，特别是对称群及其双盖，通过利用与李论的各种联系。这些联系出现在许多不同的层次上，组合的、代数的和几何的，并且涉及代数群、超群、量子群和无限维李代数的表示理论。本文拟利用顶点算子、对称群的Broue猜想，研究仿射kac - moody代数上最高权模上的Shapovalov形式，并进一步挖掘表征理论中分支规则与晶体图之间的关系。这个项目是在数学领域称为表示理论。数学工具提供了一种精确的方法来描述事物的对称性。表征理论是研究这种对称性在现实世界中产生的方式，因此它在数学、物理和化学的许多领域都有应用。在过去的几年里，我们对表征理论的理解取得了一些重大进展，这在一定程度上要归功于来自数学物理的新思想的涌入。这个项目部分涉及到所有有限群中最重要的——对称群的表示理论，它与对称函数理论密切相关。这项工作有望应用于其他数学领域，并在数学物理、统计力学和编码理论方面产生更广泛的影响。

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.