Classical Representation Theory and Categorification

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

Groups are mathematical objects arising in the study of symmetry. This project is concerned with some of the most important, fundamental and universal examples of groups: symmetric groups arising as symmetries of finite sets, and general linear groups arising as symmetries of vector spaces. Representation theory studies groups via their actions on other mathematical objects, such as vector spaces. Rather informally, representations of a group are snap-shots of the group taken from different angles. In the past several years, the idea of categorification has become important and has led to the development of higher representation theory. This involves studying actions of groups on higher mathematical structures such as categories, analyzing not only the relations between these structures (functors) but also relations between the relations (natural transformations). In particular, quiver Hecke superalgebras encode higher symmetries underlying representation theory of objects including symmetric groups, their double covers, and general linear groups. This project will develop further the theory of these and other superalgebras and apply it to improve our understanding of classical representation theory. The research in this project has potential future impact in theoretical physics and computer science. More directly this project will have an educational impact through the training of graduate students and the mentoring of young researchers in this active area. In more detail, this project is concerned with a variety of projects in representation theory of Lie algebras, finite groups, and related objects, for example Hecke algebras, quantum groups, Schur algebras and quiver Hecke superalgebras. The PI will draw on recent advances in higher representation theory, with various diagrammatically defined monoidal (super)categories playing a prominent role. On the other hand, most applications are to classical problems in representation theory such as block theory of finite groups and Schur algebras, decomposition numbers, and structure theory of finite groups. The PI will study the local description of blocks of double covers of symmetric groups up to derived equivalence, Turner-Schur (super)algebras and (super)categories and their properties, representations of quiver Hecke superalgebras and their imaginary cuspidal superalgebras, cyclotomic quiver Hecke superalgebras and their RoCK blocks, RoCK blocks of Schur superalgebras, decomposition numbers for RoCK blocks of double covers of symmetric and alternating groups and irreducible reductions modulo p for these double covers, irreducible restrictions from quasi-simple groups to subgroups and subgroup structure of finite groups. The results of the research will have applications to several 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.