Modular representations and affinizations

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.