Modular Representation Theory and Categorification with Applications

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Groups are mathematical objects arising in the study of symmetry. The main foci of this project are some of the most fundamental and universal examples of groups: symmetric groups, which arise as symmetries of finite sets, and general linear groups, which arise as symmetries of finite-dimensional vector spaces. Representation theory studies groups via their actions on other mathematical objects, such as vector spaces. Very informally, representations of a group are snap-shots of the group taken from different directions. In the past few years, the idea of categorification has become very important and has led to the development of higher representation theory. This involves actions of groups on higher mathematical structures called categories, utilizing not only the relations between these structures (functors) but also relations between the relations (natural transformations). In particular, Khovanov-Lauda-Rouquier (KLR) algebras encode higher symmetries underlying a large part of representation theory, including classical objects like symmetric and general linear groups. The goal of this project is to further build the theory of these and other algebras and apply it to improve our understanding of the classical objects of group theory. The research in this project has potential future broader impacts in computer science and theoretical physics. More directly this award will have important educational impact through the training of graduate students and mentoring young researchers in this area. In more detail, this project is concerned with several diverse projects in representation theory of Lie algebras, finite groups, and related objects such as Hecke algebras, quantum groups, Schur algebras and KLR algebras. Our perspective draws on recent advances in higher representation theory, namely categorification, with various diagrammatically defined monoidal categories and 2-categories playing a prominent role. On the other hand, many applications are to classical problems in representation theory such as block theory of finite groups and Schur algebras as well as structure theory of finite groups. We will study local description of blocks of Schur algebras up to derived equivalence, cuspidal algebras for KLR algebras, thick Heisenberg categorification, super Kac-Moody 2-categories and applications to blocks of double covers of symmetric groups, homological properties of KLR algebras, decomposition numbers, and irreducible restrictions from quasi-simple groups to subgroups. The project will have applications to other areas of mathematics including finite group theory (and its applications), Lie theory, combinatorics, representation theory, knot theory and category theory.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.