Modular representations and affinizations

模块化表示和关联

基本信息

批准号：
2302147
负责人：
Raphael Rouquier
金额：
$ 67万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2023
资助国家：
美国
起止时间：
2023-07-01 至 2028-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302147&HistoricalAwards=false
关键词：
Modular representations affinizations

项目摘要

Symmetries play a key role in various parts of science and their systematic study in various dimensions is the subject of representation theory in mathematics. In recent years it has become apparent that, more generally, it is important to investigate the phenomenon that groups of symmetries themselves possess symmetries, which leads to the notion of higher representation theory. This project will develop this viewpoint of higher representation theory further, with an emphasis on applications to solving open problems in the theory of modular representations of finite groups of Lie type. The project will provide research training opportunities for graduate students.In more detail, this project will bring higher representations of toroidal Lie algebras into the study of modular representations of finite groups of Lie type, providing two-variable conjectural decomposition matrices for those groups. An important part of the project is the development of an affinization of the theory of two-representations of Kac-Moody algebras. A second part of the project is based on a new degeneration of modular representations of finite groups of Lie type. This degeneration leads to connections with Hilbert schemes of points on surfaces and to two-variable combinatorics, which arise from perversity properties of derived equivalences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对称性在科学的各个方面起着关键作用，其在各个方面的系统研究是数学中表示理论的主题。近年来，很明显，更普遍的是，重要的是要研究对称群体本身具有对称性的现象，从而导致更高代表理论的概念。该项目将进一步发展这种高度代表理论的观点，重点是在解决有限型类型的模块化表示理论中的开放问题上的应用。该项目将为研究生提供研究培训机会。此外，该项目将使图式谎言代数的更高表示形式用于研究有限型谎言类型的模块化表示，为这些组提供了两变量的猜想分解矩阵。该项目的一个重要部分是开发Kac-Moody代数的两种代表理论的影响。该项目的第二部分是基于有限的谎言类型组模块化表示的新变性。这种退化导致与表面上的希尔伯特（Hilbert）方案以及与两变量组合物的相关性，这是由派生的等效性的变态性能引起的。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来支持的，这是值得的。