Modular representations and affinizations

对称性在科学的各个方面都起着关键的作用，在各个维度上系统地研究对称性是数学表征理论的主题。近年来，很明显，更广泛地研究对称群本身具有对称性的现象是重要的，这导致了更高表示理论的概念。本项目将进一步发展这一高等表示理论的观点，重点是应用于解决李型有限群的模表示理论中的公开问题。该项目将为研究生提供研究培训的机会。更详细地，该项目将把环状李代数的更高表示引入到李型有限群的模表示的研究中，为这些群提供二变量猜想分解矩阵。该项目的一个重要部分是Kac-Moody代数的两表示理论的仿射的发展。项目的第二部分是基于李型有限群的模表示的一种新的退化。这种退化导致了与曲面上点的希尔伯特方案和双变量组合学的联系，这是由衍生等价物的变态性质引起的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.