Titles and abstracts
Chris Bowman: Graded Temperley—Lieb algebras in categorification
In this talk we survey some recent research on (boundaried) Temperley—Lieb algebras, using ideas from KLR and Soergel bimodule diagrammatics.
Arthur Chong: Restriction of 2-representations of Soergel bimodules in Type A
Induction and restriction of representations are fundamental to the study of representation theory. This talk will briefly introduce 2-representations and give an example of restricting 2-representations of Soergel bimodules in type A.
Maud deVisscher: Graded representations of Temperley-Lieb algebras
In this talk I will discuss the representations of ordinary Temperley-Lieb algebras, one-boundary TL algebras (also known as blob algebras) and two-boundary TL algebras (also known as symplectic blob algebras). Plaza and Ryom-Hansen obtained a grading on the ordinary and the one-boundary TL algebras by relating these to quiver Hecke algebras (introduced by Khovanov-Lauda and Rouquier). The corresponding graded decomposition numbers are given by Kazhdan-Lusztig polynomials. I will review these results and then explain how we can obtain a grading for the two-boundary case using the orientifold quiver Hecke algebra introduced by Varagnolo-Vasserot and formulate a conjecture for the graded decomposition numbers.
This is joint work with Chris Bowman, Zajj Daugherty, Rob Muth and Loic Poulain D’Andecy.
Johannes Flake: From diagrammatic calculi to abelian tensor categories
Diagrammatic descriptions provide valuable insight and efficient proof techniques in representation theory. However, constructing an abelian tensor category from a given diagrammatic presentation can be challenging. Technically, this amounts to finding an abelian envelope, a problem that plays a key role in several recent developments in the field. I will explain some solutions to this problem using monoidal adjunctions. This talk is based on joint work with Jonathan Gruber, Thorsten Heidersdorf, David Hull, Robert Laugwitz, Peter Mader, and Sebastian Posur.
Jack Gidney: Normal Coverings of the Finite Classical Groups
The normal covering number of a finite group is the smallest number of proper subgroups whose union of conjugates cover the group. By a theorem of Jordan, this number must be at least two for any finite group. Building on the work of Britnell and Maróti on the linear groups, I am investigating bounds on the normal covering number of the symplectic, orthogonal and unitary groups.
Duncan Laurie: Combinatorial modules for quantum toroidal algebras
Quantum toroidal algebras are the ‘double affine’ objects within the quantum world, and possess a rich and growing representation theory. For instance, they act naturally on a range of geometric and combinatorial spaces, including the equivariant K-theory of quiver varieties and various sets of coloured partitions.
Building explicit combinatorial models for their representations is an important problem. Indeed, such descriptions clarify the structure of submodules, quotients and tensor products, as well as relate to geometric fixed-point bases. However, all existing constructions require a strict ‘tameness’ assumption, greatly restricting both the number and complexity of representations that may be treated.
After briefly introducing quantum toroidal algebras, I shall explain why this limitation arises and outline a way to overcome it using a fusion product of David Hernandez. In particular, we will build the first non-tame combinatorial modules, as well as the first examples outside type A, towards defining concrete ‘Fock modules’ in all simply-laced types. This is joint work in progress with Jiakang Bao (Tokyo, Seoul).
Ben Mills: Categorifications of the Type D Temperley-Lieb Algebra
In this talk, we present approaches to categorifying the Type D Temperley-Lieb algebra. We will discuss how the methods used in Type A can and sometimes cannot be adapted to the Type D setting, and explore the specific difficulties that arise when doing so.
Arshia Tajlili Moghanjoghi: Generalised spin Calogero Moser systems and local trivial monodromy
In [Chalykh, Feigin, Veselov 1999] the first understanding of trivial monodromy for the Schrodinger equation describing the Calogero Moser system was established. Later in [Chalykh, Goncherenko, Veselov, 1999] this understanding was extended to certain set ups where the operator related to the system is acting on a matrix valued function. Recently in [Feigin, Vasilev, Vrebec, 2026] the idea of the generalised spin Calogero Moser system was introduced. In the talk I will consolidate these three papers by introducing the generalised spin Calogero Moser system with a focus on the A type systems, discuss the trivial monodromy in the [CGV] sense and state some results relating the two papers.
Benjamin Morris: Interpolating between Temperley-Lieb and Brauer representation theory: generalised roots of unity
In 2019, Kadar, Martin, and Yu, introduced a discrete family of diagram categories/algebras with the aim of describing inhomogeneity in lattice models via the transfer matrix algebra approach. Their construction introduces the notion of “height” which is a topological condition on the crossings of a Brauer diagram. The result is a family of nested algebras interpolating between the Temperley-Lieb (TL) and Brauer algebras, setting up a fascinating program in representation theory. In this talk, I will review the construction of these, “Kadar-Martin-Yu” (KMY) algebras, and present the first fruits of the latter program. A key result on semisimplicity establishes an analogy between the TL and KMY algebras. Namely, for each integer partition, there exists a Chebyshev series of polynomials which control (some) semisimplicity criteria for the KMY algebras; this can be seen as a direct generalisation of the classical “root of unity” criteria for TL algebras. I will also present some low rank calculations as evidence that the geometric understanding of the non-semisimple, TL and Brauer cases, can be extended to the intermediate, KMY cases.
Bárbara Muniz: Tableau Algebras
We will introduce an algebra generated by semistandard Young tableaux with a simple operation — row concatenation. Using a minimal example, we will illustrate its connection to partial flag varieties and discuss how it can be used to enumerate the corresponding Plücker relations.
This is based on joint work with Spencer Daugherty, Nicolle González, Pablo S. Ocal, Jianping Pan and Jacinta Torres.
Lucia Noelle: Fock Spaces and Category O of rational Cherednik algebras
A brief introduction to Category O for the cyclotomic rational Cherednik algebra and its relationship to Fock spaces.
Theresa Ortscheidt: The Pieri Rule in the Fusion Ring
Littlewood–Richardson coefficients are structure constants of the ring of symmetric functions with respect to the Schur basis. These coefficients are non-negative and we have explicit combinatorial formula for computing them.
We are interested in the fusion ring H(k,ℓ), first introduced by Gepner in 1991, which is related to certain representations of type A Hecke algebras at roots of unity. Goodman and Wenzl showed that the fusion ring can be viewed as a quotient of the ring of symmetric functions, with a basis given by Schur polynomials indexed by so-called (k, ℓ) – diagrams. The fusion coefficients with respect to this basis are known to be non-negative as well, but at the moment, no manifestly positive formula exists.
In this talk I will show an example for computing fusion coefficients for the special case when one of the factors is indexed by a single row, using a solvable lattice model.
Willoughby Seago: Marked Tableaux
Young tableaux are common in many areas of representation theory. In this talk I will discuss a generalisation, known as marked tableaux, which appear in similar contexts where supersymmetry is involved. Much that is known about Young tableaux generalises to these marked tableaux, including the RSK algorithm, crystal structures, and lattice models.
Charles Senécal: Clasps in sl_3 webs at roots of unity
The sl_3 web category is a combinatorial model for the representation theory of the quantum group U_q(sl_3). We study the category from the cellular point of view, exploiting diagrammatic methods and light-ladder bases to compute branching rules, homomorphisms, and composition factors of cell modules when q is a root of unity.
We then apply these results to the study of clasps. In the semisimple case, these higher rank analogues of the Jones—Wenzl idempotents can be constructed using Elias’s recursive triple-clasp expansion. When q is a root of unity, however, their construction is considerably more subtle and we use the aforementioned results from cellularity to obtain formulas for some of the non-semisimple clasps.
Sasha Shapiro: Coisotropic reduction on quantum cluster varieties
Coisotropic reduction of an associative algebra A by its 1-sided ideal I is the quotient N(I)/I of the normalizer of I by the ideal. If A has a structure of a quantum cluster algebra, one may ask under which conditions on the ideal I, the quotient N(I)/I inherits a cluster structure from A. While the question seems hopeless in this generality, I will demonstrate a class of examples for which N(I)/I indeed admits a cluster structure. I will also discuss a powerful application these examples yield in quantum Teichmuller theory and how the double affine Hecke algebra appears in this context. This talk will be based on a joint work with Corey Lunsford and Gus Schrader.
Lei Shi: Centers and cocenters of cyclotomic KLR algebras
Cyclotomic KLR algebras play an important role in the categorification of integrable highest weight representations. In this talk, I will discuss some recent results on their centers and cocenters, including connections with the corresponding highest weight representations. I will also mention their connections with the cohomology of quiver varieties.
Jasper Stokman: Discrete quantum walks, Hecke algebras and Hall-Littlewood polynomials
Quantum walks are quantum mechanical versions of random walks on graphs. A well-known example of a discrete-time quantum walk is the Grover walk, which generalizes Grover’s celebrated quantum algorithm for unstructured database searches.
The transition operator of a discrete quantum walk is the composition of two involutions: the first involution represents the flipping of a quantum coin, the second represents the quantum step. The abstract group generated by two involutions is the infinite dihedral group, which is an example of an affine Weyl group. Group algebras of affine Weyl groups admit natural flat deformations called affine Hecke algebras, which play an important role in representation theory of reductive p-adic groups.
In the first part of the talk I will give a short introduction to discrete quantum walks from the perspective of representation theory of the infinite dihedral group. In the second part of the talk I will introduce two-parameter generalisations of discrete quantum walks by replacing the role of the infinite dihedral group by its affine Hecke algebra. Finally, I will explain how perfect state transfer in these models can be described in terms of Hall-Littlewood polynomials.
James Timmins: Coherence of rings in representation theory
Coherent rings are a class of noncommutative rings which generalise Noetherian rings. In this talk I'll introduce them, explain their relevance to representation theory, and describe what is known about coherence of some significant examples, including universal enveloping algebras.
Lewis Topley: Harish-Chandra bimodules in positive characteristics
I will report on a joint work with Lewis Groves and Matt Westaway, soon to appear. Let g = Lie(G) be the Lie algebra of a simple algebraic group over an algebraically closed field k. A Harish-Chandra bimodule is a U(g)-U(g)-bimodule M with an action of G such that the adjoint g-action on M integrates to G. In 1980 Bernstein and Gelfand studied the category of such modules when char(k) = 0: they classified the simple objects and showed that the subcategory with a fixed generalised central character is equivalent to a block of category O for g. Their work later inspired Soergel to introduce his ubiquitous category of Soergel bimodules. More recently, HC bimodules have featured in works of Losev and Bezrukavnikov—Riche in the case char(k) = p > 0. In our work we classify the simple objects in the category, when p > 0 is very good for G.
Angelina Vargulevich: On reciprocal characters and quantum affine Schur–Weyl duality
Quantum affine Schur–Weyl duality, due to Chari and Pressley, relates finite-dimensional representations of affine Hecke algebras of type GL to those of quantum affine algebras of type A. It is natural to ask how invariants of representations on the two sides match under this duality.
For quantum affine algebras there is a natural notion of character, the q-character of Frenkel and Reshetikhin, which records the dimensions of certain weight spaces. In joint work with Maxim Gurevich, we construct a new invariant on the affine Hecke algebra side, which we call the reciprocal character. Under quantum affine Schur–Weyl duality it corresponds to the dominant part of the q-character. It also admits a natural interpretation in the representation theory of p-adic groups.
If time allows, I will talk about an interpretation of this result in another setting: the dominant part of the q-character corresponds to monomial bases in Lusztig's algebra, through Ariki's categorification.
Scott Warrander: Bar involutions and canonical bases in K-theory via stable envelopes
Many modules over algebras with canonical basis theories, such as affine Hecke algebras and affine quantum groups, can be realised using equivariant K-theory. In the late 90s, Lusztig began studying these K-theory groups and trying to write down a geometric definition of their bases, characterising them by invariance under an appropriately defined bar involution. In this short talk I will discuss a new approach to defining bar involutions due to Hikita, using stable envelopes.
Milen Yakimov: Quantum symmetric pairs via short star products and dual quantum Popov degenerations
The goal of the talk is to describe a new approach to the theory of quantum symmetric spaces based on the notions of short star products of Beem-Peelaers-Rastelli and Etingof-Stryker and a quantum version of Popov degeneration. The central result is that every quantum symmetric subalgebra is realized via a short star product on a quantum horospherical subalgebra by a dual quantum Popov degeneration. We prove that the latter are always short star products and use it to give short and conceptual proofs of many results for quantum symmetric pairs that before were proved by long and diverse methods. The methods also show that all quasi K-matrices for these quantum Kac-Moody symmetric pairs are explicitly expressible in terms of the much studied quasi R-matrices of Drinfeld and Lusztig. This is a joint work with Stefan Kolb.