# Graded representations of symmetric groups and related algebras

SPEAKER: Jonathan Brundan

TITLE: Graded representations of symmetric groups and related algebras

ABSTRACT I want to talk about some new algebras that have appeared in recent works of Khovanov and Lauda. We called them Khovanov-Lauda algebras at first. But it turns out the same algebras were discovered independently by Rouquier (unpublished), who calls them QUIVER HECKE ALGEBRAS. Our main theorem (this is all joint with Sasha K) shows these algebras attached to linear or cyclic quivers are isomorphic to certain cyclotomic Hecke algebras. As a special case these algebras are isomorphic to group algebras of symmetric groups. This is interesting because quiver Hecke algebras are naturally graded, so this leads to a remarkable hidden grading on the group algebras of symmetric groups. I’ll try also to explain a natural occurence of these quiver Hecke algebras in type A which explains the hidden grading (as a shadow of some Koszulity) and discuss some related categorification conjectures.