Hidden Gradings in Representation Theory

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

This project is concerned with some diverse projects in representation theory of Lie algebras, finite groups, and related objects such as Hecke algebras, quantum groups, Khovanov-Lauda-Rouquier (KLR) algebras, Brauer algebras, W-algebras, Yangians, and Lie superalgebras. One of the unifying ideas underpinning the proposal is the study of hidden gradings ubiquitous in many important situations in representation theory, as well as a related idea of categorification of quantum objects. Several of the projects are concerned with the representation theory of KLR algebras, which were introduced to categorify quantum groups. In particular the PIs will initiate a study of homological algebra of KLR algebras of finite type, and study a new phenomenon of imaginary Schur-Weyl duality arising from KLR algebras of affine type. Completion of these projects will lead to substantially better understanding of these algebras both in characteristic zero and in positive characteristic. Other projects are concerned with graded category O for the general linear Lie superalgebra, Deligne's category Rep(GL_delta), graded representations of spin symmetric groups, and the Aschbacher-Scott program of classifying maximal subgroups in finite classical groups. The proposal is expected to have applications to several other areas of mathematics including finite group theory (and its applications), Lie theory, combinatorics, representation theory, knot theory and category theory. Representation theory is a core topic in pure mathematics, with many connections to other areas of mathematics, mathematical physics, computer science, chemistry and even biology. In the last few years the subject has been influenced heavily by ideas from higher category theory, leading to the introduction by Khovanov, Lauda and Rouquier of some remarkable new structures known as KLR algebras. These algebras encode higher symmetries underlying a large part of combinatorial representation theory, including classical objects like symmetric and general linear groups. The goal of the project is to build further the theory of these algebras and apply it to improve our understanding of these classical objects. The basic 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 the on-going efforts of both PIs in mentoring other young researchers in this area. The award will indirectly support the promotion of knowledge of the methods and results of this beautiful subject area both nationally and internationally, through the active involvement of both PIs as organizers of major conferences and as editors of leading specialist journals.