Representations of Finite Groups and Algebraic Lie Theory

有限群的表示和代数李理论

• 批准号: 0139019
• 负责人: Alexander Kleshchev
• 金额: $ 35.7万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0139019&HistoricalAwards=false
• 关键词: Representations; Finite; Groups; Algebraic; Lie

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.

研究人员将通过利用与谎言理论的各种联系，继续他们对有限群体的表示理论，尤其是对称群体及其双重覆盖的研究。这些连接出现了不同级别的不同级别，组合，代数和几何，涉及代数群，超组，量子组和无限维度谎言代数的表示理论。研究人员打算特别研究Shapovalov形式，通过顶点操作员，Broue对thesymmetric群体的猜想，在Agginekac-Moody代数上的最高权重模块上进行研究，并进一步利用了代表理论中的分支图和晶体图之间的关系。 数学工具提供了描述某物对称性的精确方法。 代表理论是对这种对称性在现实世界中可能出现的方式的研究，因此它在许多数学，物理和化学领域都有应用。 在过去的几年中，我们对表示理论的理解有了一些重大进展，这要归功于数学物理学的新思想的新涌入。该项目与所有这些群体中最重要的对称群体中最重要的代表理论联系在一起，该理论与对称函数的理论密切相关。这项工作有望在其他数学领域应用，并对数学物理学，统计力学和编码理论产生更大的影响。