Tag - Soergel bimodules

Scott Larson: A Categorification of the Lusztig-Vogan Module

4th April, 2022

Continuous actions of real reductive groups are often studied by first linearizing the action to spaces related to functions, then using algebra via Lie algebras and compact groups (cf. Gelfand, Harish-Chandra, Vogan). This paradigm essentially simplifies to the easier problem of studying a complex algebraic group K acting on flag varieties. K-orbit closures are important for representation theory, are generalizations of Schubert varieties, and certain properties are explicitly determined via equivariant resolutions of singularities. In joint work with Anna Romanov, we provide a geometric and algebraic categorification of the Lusztig-Vogan module using the equivariant derived category. Our methods allow us to compute cohomology of all fibres of resolutions constructed quite generally and generalize Soergel bimodule techniques from complex to real reductive algebraic groups.

Geordie Williamson: Spectra in representation theory

18th June, 2021

In geometric representation theory cohomology, intersection cohomology and constructible sheaves show up everywhere. This might seem strange to an algebraic topologist, who might ask: why this emphasis on cohomology, when there are so many other interesting cohomology theories (like K-theory, elliptic cohomology, complex cobordism, ...) out there? They might also ask: is there something like "intersection K-theory", or "intersection complex cobordism"? This is something I've often wondered about. I will describe work in progress with Ben Elias, where we use Soergel bimodules to investigate what KU-modules look like on the affine Grassmannian. We have checked by hand that in types A1, A2 and B2, one gets something roughly resembling the quantum group. Speaking very roughly, the intersection K-theory of Schubert varieties in the affine Grassmannian should recover the irreducible representations of the quantum group. Inspirations for this work include a strange Cartan matrix discovered by Ben Elias, and work of Cautis-Kamnitzer.

Anna Romanov: A categorification of the Lusztig-Vogan module

26th April, 2021

Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lusztig-Vogan in the 80s in terms of a geometrically defined module over the associated Hecke algebra. In this talk, I'll describe a categorification of this module using Soergel bimodules, with a focus on examples.

Nicolle Gonzalez: A skein-theoretic Carlsson-Mellit algebra

2nd November, 2020

The shuffle conjecture was a big open problem in algebraic combinatorics which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This conjecture was finally solved after 14 years by Carlsson and Mellit by the introduction of a new interesting algebra denoted Aq,t. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. In later work by Carlsson, Mellit, and Gorsky this algebra and its representation was realized using parabolic flag Hilbert schemes and was also shown to contain the generators of the elliptic Hall algebra. I will discuss a new topological formulation of Aq,t and its representation over a thickened annulus and a categorification thereof over the derived trace of the Soergel category. This is joint work with Matt Hogancamp.

Eugene Gorsky: Link homology and Hilbert schemes

28th September, 2020

The category of Soergel bimodules categorifies the Hecke algebra. Khovanov and Rozansky used Soergel bimodules to define triply graded link homology (also known as Khovanov-Rozansky homology) which categorifies HOMFLY-PT link invariant. I will survey some results and conjectures relating Soergel bimodules and Khovanov-Rozansky homology with the Hilbert scheme of points on the plane. The talk is based on joint works with Matt Hogancamp, Andrei Negut, Jake Rasmussen and Paul Wedrich.