talks & conferences

2026

Tensor categories in representation theory & vice versa Uppsala University, Uppsala, Sweden August 24–27, 2026 · Conference

Circular Leaves for the diagrammatic Hecke category in type $A_3$

The diagrammatic Hecke category is a diagrammatic version of the category of Soergel bimodules, which categorifies the Hecke algebra. All of its morphisms are described by a small number of generators and relations, and a basis of every Hom space is known: Libedinsky’s double leaves. However, so far there is no implementation on a computer. We present new work that makes such an implementation possible. Concretely, we describe circular leaves for any word. They are the most simple diagrams that one can construct out of double leaves. Simple in the sense that local moves which make diagrams contain fewer nodes lead to circular leaves. We conjecture that they are in a 1:1 correspondence with double leaves and give a concrete algorithm for rewriting any diagram as a linear combination of circular leaves. We use this to prove the Zamolodchikov relation in type $A_3$, and propose a way to find the missing relation in type $H_3$.

Seminar der Arbeitsgruppen Diskrete Mathematik/Geometrie & Diskrete Geometrie TU Berlin, Germany August 5, 2026 · Seminar

Circular double leaves in the diagrammatic Hecke category

The diagrammatic Hecke category by Elias–Williamson categorifies the category of Bott–Samelson bimodules associated to an arbitrary Coxeter group. While diagrams often are very helpful to replace complicated algebraic computations, and therefore have been used extensively to study Soergel bimodules, they have not yet been implemented into a computer. This is because no algorithm was known that can verify whether two diagrams are actually the same, even though due to Libedinsky it is completely known how to construct a diagrammatic basis of any Hom-space. We conjecture a normal form for diagrams in the case $A_3$ and present a new diagrammatical software package. Concretely, we first define the diagrammatic Hecke category with generators and relations; explain the so-called basis of double leaves by Libedinsky; and show which local rules we use to turn any diagram into a sum of “circular leaves”. Furthermore we present how the computer implementation works and how anyone can play with these diagrams.

Event

Oberseminar Darstellungstheorie Max Planck Institute for Mathematics (MPIM), Bonn, Germany June 26, 2026 · Seminar

The Zamolodchikov relation

The Hecke algebra is categorified in two ways. Firstly algebraically by Soergel bimodules, secondly diagrammatically by the diagrammatic Hecke category. The latter category is generated by strands in one color for every generating reflection of the underlying Coxeter group, subject to certain one-color, two-color and three-color relations. While in type $A_3$ and $B_3$ Elias–Williamson gave an exact description of the three-color (also called Zamolodchikov) relation, there is still a gap in type $H_3$. Each diagrammatic relation is a certain equality of two morphisms between Bott-Samelsons. We will explain how one can go in-between the algebraic and diagrammatic world to compute how these morphisms factor over all indecomposable summands of these Bott-Samelsons. For this we need to explain the construction of idempotents inside the diagrammatic category. As a consequence we can 1. describe a fake Zamolodchikov relation in $A_3$, 2. analyse all possible variants of Zamolodchikov relations in type $A_3$ and $B_3$, and 3. describe how one could find the missing gap in $H_3$, as well as showing how far computer calculations already came.

Event

International Workshop on quantum groups, categorification and related topics Tianyuan Mathematical Research Center, Kunming, Yunnan, China June 07–12, 2026 · Conference

The Zamolodchikov relation

The diagrammatic Hecke category categorifies the category of Soergel bimodules. For every generating reflection of the underlying Coxeter group we have one color of strands in the diagrammatic category. They satisfy certain one-color, two-color and three-color relations. The latter is also called Zamolodchikov relation. While in type $A_3$ and $B_3$ Elias–Williamson gave an exact description there is still a gap in type $H_3$. We will explain how a solution can look like; what happens in type $A_3$ and $B_3$ for variants of the original Zamolodchikov and how far the calculation in type $H_3$ has come.

Event schedule

Mini-conference: Categorification University of East Anglia, Norwich, UK January 12–13, 2026 · Conference

Idempotents and dimensions in the asymptotic Hecke category

The diagrammatic Hecke category, introduced by Elias–Williamson, provides a categorical framework for studying Soergel bimodules through diagrams. We present results from recent joint work with Ben Elias and Dani Tubbenhauer. Our main contributions includes an algorithmic construction of idempotents, as well as the computations of the dimensions of objects inside the asymptotic Hecke category. We show these with many diagrammatic examples.

2025

Joint Block Seminar on Category Theory University of Zurich, Switzerland December 2, 2025 · Seminar

Computing Idempotents and Dimensions in the Asymptotic Hecke Category

Event

Annual Meeting 2025 of the SFB TRR 195 University of Tübingen, Germany September 22–25, 2025 · Conference

Computing in the asymptotic Hecke category

Event

2023

Block Seminar, SFB 195 Graduate Programme Trippstadt, Germany September 18–22, 2023 · Organized event

An Introduction to Categorification

Event