The Zamolodchikov relationOberseminar Darstellungstheorie

by Liam Rogel (University Kaiserslautern–Landau)

Friday Jun 26, 2026, 2:15 PM → 3:15 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Seminar Room (Max Planck Institute for Mathematics)

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. descibe how one could find the missing gap in $H_3$, as well as showing how far computer calculations already came.