Cyclotomic KLR algebras of affine type COberseminar Darstellungstheorie

by Andrew Mathas (University of Sydney)

Thursday Jul 3, 2025, 10:00 AM → 11:00 AM Europe/Berlin

Endenicher Allee 60/0.007 - Seminarraum (Mathezentrum)

The Khovanov–Lauda–Rouquier (KLR) algebras are a family of graded algebras indexed by quivers. For symmetrisable quivers, these algebras are important because they categorify the positive part of the corresponding quantum group. These algebras admit natural cyclotomic quotients, indexed by dominant weights, which categorify highest weight modules.

Over a field, Brundan and Kleshchev proved that the cyclotomic KLR algebras of affine type A are isomorphic to cyclotomic Hecke algebras of type A. These include the group algebras of symmetric groups, their Iwahori–Hecke algebras, and the (degenerate) Ariki–Koike algebras. For other types, the KLR algebras are genuinely new objects that were not studied prior to the KLR epoch.

This talk gives a gentle introduction to the cyclotomic KLR algebras of affine type C. I will discuss what is currently known about their structure and representation theory, as well as open questions. In particular, there are striking parallels with the classical representation theory of the symmetric groups, but also important differences where the theories diverge, and is far from understood.