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Organizing Committee
University of Oregon
UC Berkeley
Columbia University
University of New South Wales
Abstract
Many classical results connecting representation theory and combinatorics can be succinctly understood through the concept of categorification. Two celebrated examples are Frobenius’s 1900 theorem showing that the ring of symmetric functions is categorified by the representation theory of symmetric groups, and the Kazhdan-Lusztig conjecture which relates category O for a complex semisimple Lie algebra to the Hecke algebra of the corresponding Weyl group. In these and other examples, positivity properties observed at a combinatorial level are explained conceptually by the existence of a categorification. The original proofs of the Kazhdan-Lusztig conjecture found in 1981 were geometric in nature, involving perverse sheaves and D-modules on the underlying flag varieties. Elias and Williamson found an alternative, purely algebraic proof in 2014, exploiting a diagrammatic description of the monoidal category of Soergel bimodules.
The diagrammatic approach has its origins in the discovery of new quantum invariants of knots and links in the 1980s. Crane and Frenkel raised the idea of categorifying quantum groups, hence, link invariants, already in the 1990s, and this vision prompted the development of powerful link homology theories such as Khovanov-Rozansky homology. The categorification of quantum groups involves Khovanov-Lauda-Rouquier algebras, which are often presented diagrammatically. They are a building block for Kac-Moody 2-categories, which categorify Lusztig’s modified integral form for quantized enveloping algebra. There have been many remarkable developments in this direction in the last few years, including the introduction by Webster of more general algebras categorifying tensor products, DG versions of these algebras which categorify Verma modules, and p-DG versions which are being used to categorify link invariants at roots of unity.
The field of diagrammatic categorification is still in its early stages, but it has already had a significant impact on more traditional mathematics. This workshop aims to unite both established experts and emerging scholars across various domains of diagrammatic categorification, including representation theory, combinatorics, and link homology.
The diagrammatic approach has its origins in the discovery of new quantum invariants of knots and links in the 1980s. Crane and Frenkel raised the idea of categorifying quantum groups, hence, link invariants, already in the 1990s, and this vision prompted the development of powerful link homology theories such as Khovanov-Rozansky homology. The categorification of quantum groups involves Khovanov-Lauda-Rouquier algebras, which are often presented diagrammatically. They are a building block for Kac-Moody 2-categories, which categorify Lusztig’s modified integral form for quantized enveloping algebra. There have been many remarkable developments in this direction in the last few years, including the introduction by Webster of more general algebras categorifying tensor products, DG versions of these algebras which categorify Verma modules, and p-DG versions which are being used to categorify link invariants at roots of unity.
The field of diagrammatic categorification is still in its early stages, but it has already had a significant impact on more traditional mathematics. This workshop aims to unite both established experts and emerging scholars across various domains of diagrammatic categorification, including representation theory, combinatorics, and link homology.
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Workshop Schedule
Monday, October 20, 2025
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:50 AM EDT
Quantum loop algebras, simple modules, and q-characters
11th Floor Lecture Hall
Speaker
Andrei Negut, EPFL
Session Chair
Anna Romanov, University of New South Wales
Abstract
I will survey recent developments on the construction of simple modules of quantum loop algebras (of arbitrary Kac-Moody type) with an eye toward categorification.
10:00 - 10:20 AM EDT
Coffee Break
11th Floor Collaborative Space
10:20 - 11:10 AM EDT
Elliptic Hall algebra and its categorifications
11th Floor Lecture Hall
Speaker
Eugene Gorsky, UC Davis
Session Chair
Anna Romanov, University of New South Wales
Abstract
I will review various presentations of the elliptic Hall algebra by generators and relations, including the recently discovered relation with the double Dyck path algebra of Carlsson and Mellit. I will then explain how these generators appear in various categorical setups such as the derived category of the commuting stack and the trace of the affine Hecke category, and state several conjectures about them. The talk is based on joint works with Andrei Negut, Nicolle Gonzalez and Jose Simental.
11:30 - 11:35 AM EDT
Soergel algebras as bound-quiver algebras: the infinite dihedral case
Lightning Talks - 11th Floor Lecture Hall
Speaker
Theo Pinet, McGill University
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
The blob algebra TLbN is a finite-dimensional cellular quotient of the Hecke algebra of type B. It appears naturally in statistical physics and admits a notoriously intricate representation theory.
In this talk, we use Soergel bimodules of affine type A1 to give a bound-quiver realization for blocks of TLbN. We then use this novel realization to deduce surprising results about the endomorphism algebras of projective TLbN-modules.
The talk is based on joint work with Alexis Leroux-Lapierre and Yvan Saint-Aubin.
11:35 - 11:40 AM EDT
Highest weight theory for representations of general linear groups in Verlinde cateories.
Lightning Talks - 11th Floor Lecture Hall
Speaker
Aleksandra Utiralova, University of Oregon
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
Verlinde category Ver_p serves as an analog of super vector spaces for Tannakian reconstruction of tensor categories in characteristic p. This means that any nice enough tensor category (Frobenius exact, of moderate growth) linear over a field of char p is equivalent to the category of representations of some affine group scheme in Ver_p.
I will talk about the classification of representations of GL(X) for X an object in Ver_p and the highest weight theory for them.
11:40 - 11:45 AM EDT
A diagrammatic approach to reflection functors
Lightning Talks - 11th Floor Lecture Hall
Speaker
Haruto Murata, the University of Tokyo
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
For an arbitrary symmetrizable generalized Cartan matrix, the category of modules over the KLR algebra (Quiver Hecke algebra) provides a categorification of the associated quantum group. In this talk, I will present a construction of reflection functors that categorify Lusztig’s braid group action, from the viewpoint of 2-representation theory. Similar functors have recently been constructed independently, by Kashiwara-Kim-Oh-Park, using a different approach.
11:45 - 11:50 AM EDT
Higher idempotent completion for some diagrammatic categories
Lightning Talks - 11th Floor Lecture Hall
Speaker
Isabela Recio, Universität Hamburg
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
What is an good notion of an idempotent map inside a 2-category? What would a categorified notion of idempotent completion look like? Following constructions introduced by Reutter-Douglas and Gaiotto-Johnson--Freyd, we explore some of the applications of higher idempotent completion in the case of Soergel bimodules, relying heavily on their diagrammatic incarnation, and the deformed foam categories underlying combinatorial formulations of deformed Khovanov-Rozansky homology.
11:50 - 11:55 AM EDT
Khovanov-Rozansky Homology, Affine Springer Fibers, and q,t-Catalans
Lightning Talks - 11th Floor Lecture Hall
Speaker
Joshua Turner, UC Davis
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
We’ll briefly talk about the relationship between homology of affine Springer fibers and Khovanov-Rozansky homology of links, and the q,t-Catalan combinatorics that arise from both.
11:55 AM - 12:00 PM EDT
Action of the Witt algebra on categorified quantum groups
Lightning Talks - 11th Floor Lecture Hall
Speaker
Jernej Grlj, University of Southern California
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
In a recent paper, joint with A. Lauda, we constructed an action of the positive half of the Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. We will present this action. We will specialize the action to the type A case. In this case the presented action induces an action of the positive Witt algebra on \mathfrak{gl}_n-foams, recovering the action of Qi, Robert, Sussan, and Wagner.
12:00 - 12:05 PM EDT
2-Categorifical affine symmetries and quantum boson algebras
Lightning Talks - 11th Floor Lecture Hall
Speaker
Sam Qunell, UCLA
Session Chair
J. Elisenda Grigsby, Boston College
Abstract
Representations of KLR (quiver Hecke) algebras categorify the positive part of the quantum group associated to any symmetrizable Cartan matrix. This categorical perspective makes certain symmetries more natural to study. For example, the induction and restriction functors between categories of KLR algebra modules play an important role in the theory. A closer investigation of these functors reveals surprising new symmetries. In this talk, I explain how the induction and restriction functors for KLR algebras can be used to obtain a 2-representation of the corresponding affine positive part in type A. I also describe a new categorification of a closely related algebra, the q-boson algebra, in all symmetrizable Kac-Moody types.
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