Prof. Alexander Kleshchev
“Representation Theory of Symmetric Groups and Categorification”
University of Oregon and Institute for Advanced Studies
Abstract:
We discuss modular branching rules of symmetric groups and describe their application to Mullineux Conjecture on tensor products of irreducible representations of symmetric groups with the sign representation. We explain how these results lead naturally to the ideas of higher representation theory/categorification. In particular, we explain how module categories of symmetric groups and various Hecke algebras form 2-representations of 2-Kac-Moody categories. This involves an isomorphism of blocks of symmetric groups and Hecke algebras with Khovanov-Lauda-Rouquier algebras which is of independent interest. In particular, it reveals some non-trivial grading in representation theory of symmetric groups which is not seen classically. We discuss applications of higher representation theory to classical representation theory, in particular, to Broue’s Abelian Defect Conjecture for symmetric groups and their double covers. Broue’s Conjecture describes blocks of finite groups with abelian defect groups in terms of local subgroups up to derived equivalence. We further develop these ideas to blocks of symmetric groups of not necessarily abelian defect.
Lecture 1: Monday, January 16, 2023 at 15:30 Lecture slides
Lecture 2: Wednesday, January 18, 2023 at 15:30 Lecture slides
Lecture 3: Thursday, January 19, 2023 at 15:30 Lecture slides
Light refreshments will be given before the talks on the 5Th floor.