QUACKS conference

Ben Webster (Perimeter Institute), David Rose (University of North Carolina)

Organizer's time: 2020-08-12 08:00 America/Los_Angeles

Duration: 6 hours

Schedule at https://pages.uoregon.edu/belias/QUACKS/schedule.html

Registration at https://pages.uoregon.edu/belias/QUACKS/registration.html

Ben Webster (Perimeter Institute and University of Waterloo): Howe to translate Gelfand-Tsetlin

Soergel bimodules have natural manifestations in 3 different contexts: combinatorial (i.e. diagrammatic calculus), geometric (i.e. perverse sheaves on the flag variety) and representation theoretic (i.e. Harish-Chandra bimodules/category O).

In each of these contexts, there are generalizations that might interest you: - on the combinatorial side, there is a categorification of the kth tensor power of C^n via KLRW algebras, studied by Khovanov-Lauda-Sussan-Yonezawa in the context of "categorical symmetric Howe duality"; they propose that the action of S_k on this tensor power categories to an action of Soergel bimodules, and prove this for n=2. - on the geometric side, you can replace a flag by a sequence of maps V_1 -> V_2 -> ... -> V_m -> C^n (considered up to isomorphism) without requiring injectivity. You can convolve B-equivariant sheaves on this space X with perverse sheaves on B\G/B. - on the representation theoretic side, you can replace category O by Gelfand-Tsetlin modules (the modules over U(gl_n) which are locally finite under the Gelfand-Tsetlin subalgebra S). Like category O, these carry an action of translation functors, which are effectively a copy of Soergel bimodules.

In fact, all of these generalizations are the same! The modules over appropriate KLRW algebras are a graded lift of the category of Gelfand-Tsetlin modules, and Koszul dual to the category of B-equivariant perverse sheaves on X, and all of these equivalences are compatible with the actions of Soergel bimodules. I'll try to explain this result, and how whatever your perspective, none of the objects involved are as scary as you might think.

David Rose (University of North Carolina): Webs in type C.

Abstract: A fundamental question one can ask is for a presentation of a given algebraic object via generators and relations. The advent of quantum topology suggested considering this question in the case of categories of quantum group representations, where its resolution gives insight into link invariants and TQFT, and serves as a starting point for categorification. We will discuss results of Kuperberg in rank < 3 and Cautis-Kamnitzer-Morrison in type A, and then turn to recent work of the speaker and collaborators that give the first results on this problem for rank > 2 outside type A. This is based on joint work with Tatham, and time permitting might also touch on work in progress with Bodish, Elias, and Tatham.

Submitted by: Eugene Gorsky