Modular representations and affinizations

对称性在科学的各个部分都起着关键作用，它们在各个维度上的系统研究是数学表示论的主题。近年来，人们越来越清楚地认识到，更一般地说，研究对称群本身具有对称性的现象是很重要的，这导致了高级表示论的概念。这个项目将进一步发展这种观点的高级表示理论，重点是应用程序解决开放的问题，在理论的模表示的有限群的李型。本研究计划将为研究生提供研究训练的机会，更详细地说，本研究计划将把环形李代数的高级表示引入李型有限群的模表示的研究中，为这些群提供二元的代数分解矩阵。该项目的一个重要组成部分是发展的仿射理论的两个表示的卡茨穆迪代数。该项目的第二部分是基于一个新的退化模表示有限群的李型。这种退化导致与希尔伯特计划的点在表面上和两个变量的组合，这是从反常属性派生equivalents.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.