Representation Theory and Categorification Seminar (Spring 2025)

This is a continuation of last semester's learning/research seminar on representation theory, often with an eye towards categorification. Inside talks will likely be disconnected talks from across representation theory and categorifcation, reflective of what the speaker is currently interested in working on; in particular this seminar is not cumulative and you are welcome to attend whatever talk you find interesting. We also plan to have outside talks. There may be talks relating to affine Hecke algebras, KLRW algebras, quantum affine algebras, Soergel bimodules, etc..

Talks will roughly be 1.25-1.5 hours, e.g. 45 minutes followed by a 5 minute break, followed by 45 more minutes. Please email Fan or Alvaro at fz2326@columbia.edu or alm2297@columbia.edu if you'd like to give a talk!

Schedule

01/24
Speaker: Cailan Li (Academia Sinica)
Title: How to compute Torus Link Homology
Abstract: Triply Graded link homology (HHH) categorifies the HOMFLY polynomial but it's quite hard to compute from the definition. In this talk we go over a paper of Elias and Hogancamp in which they introduce a method to compute HHH of torus links using categorified projectors.
01/30
Speaker: N/A
Title: Organizational meeting
Abstract: Since last week was Cailan's talk, we are having our organizational meeting this week instead.
02/06
Speaker: Davis Lazowski
Title: W-actions on q-characters
Abstract: After reviewing basic definitions and examples, we explain how to define a Weyl group action on q-characters of a quantum affine algebra following Frenkel-Hernandez.
02/13
Speaker: N/A
Title: No seminar
Abstract:
02/20
Speaker: N/A
Title: No seminar
Abstract:
02/27
Speaker: N/A
Title: No seminar
Abstract:
03/06
Speaker: Felix Roz
Title: Introduction to Soergel bimodules
Abstract: I will discuss the Kazhdan-Lusztig basis for Hecke algebras and define Soergel bimodules. At the end I will state the categorification theorem. The talk will follow chapters 3-4 of Elias et al.
03/13
Speaker: Hank Chen (BIMSA)
Title: Combinatorial quantization of 4d derived Chern-Simons theory and a target for higher ribbon functors
Abstract: Procedures for producing 3d TQFTs from the data of quantum group Hopf algebras are well-known, and it is widely accepted that to lift these constructions to higher-dimensions, we need to perform a categorification. However, many unknowns and obstacles still stand in our way. As a first step toward this program, I will introduce a higher-homotopy generalization of the 3d Chern-Simons theory to 4-dimensions, and describe the higher-categorical framework in which it can be quantized on a lattice. Based on the geometry of stratified 3-manifolds and its 2-skeleton, I will show how we can extract data from the underlying 4d action in order to equip the underlying discrete degrees-of-freedom of 2-Chern-Simons holonomies with the structure of a Hopf cocategory. Then, if time permits, I will discuss the representation 2-category of its quantum gauge symmetries, and demonstrate that it is ribbon tensor (and what this means). This servers as the target for the ribbon 2-functor that determines the 4d 2-Chern-Simons TQFT via the 2-tangle hypothesis. This is based on my recent works https://arxiv.org/abs/2501.06486 and https://arxiv.org/abs/2501.08041.
03/20
Speaker: N/A
Title: Spring Break
Abstract:
03/27
Speaker: N/A
Title: Simons conference at Flatiron
Abstract:
04/03
Speaker: Fan Zhou
Title: Diagrammatic Soergel calculus
Abstract: We continue with Soergel theory. We will cover chapter 5 from the book and then give the diagrammatic description of morphisms between Bott-Samelson bimodules, colloquially known as diagrammatic Soergel calculus, or the diagrammatic Hecke category. We will touch base with (a principal central block of) category O using this theory. If time permits I will sketch a cute proof of Kostant's theorem on Lie cohomology using this diagrammatic calculus.
04/24
Speaker: David Hernandez (Université Paris Cité)
Title: Baxter polynomials for Q-operators and representations of shifted quantum groups
Abstract: We explain the application of polynomiality of Q-operators to representations of truncated shifted quantum affine algebras (and quantized K-theoretical Coulomb branches). The Q-operators are transfer-matrices associated to prefundamental representations of the Borel subalgebra of a quantum affine algebra, via the standard R-matrix construction. In a joint work with E. Frenkel, we have proved that, up to a scalar multiple, they act polynomialy on simple finite-dimensional representations of a quantum affine algebra. This establishes the existence of Baxter polynomial in a general setting (Baxter polynomiality). In the framework of the study of K-theoretical Coulomb branches, Finkelberg-Tsymbaliuk introduced remarkable new algebras, the shifted quantum affine algebras and their truncations. We propose a conjectural parameterization of simple modules of a non simply-laced truncation in terms of the Langlands dual quantum affine Lie algebra (this has various motivations, including the symplectic duality relating Coulomb branches and quiver varieties). We prove that a simple finite-dimensional representation of a shifted quantum affine algebra descends to a truncation as predicted by this conjecture. This is derived from Baxter polynomiality.

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