Description du sujet de thèse

The goal is to use Garside structures to study the faithfulness of Khovanov-Seidel braid representation for Artin-Tits groups.
Artin-Tits groups are symmetry groups associated to quantum groups, which generalize braid groups. Their algebraic and geometric study is particularly rich, and difficult in broad generality. The use of a so-called Garside structure, which allows for a preferred writing of elements and strong algorithmic properties, is very useful, but only few examples are known.
Thomas Haettel (with Jingyin Huang, Duke Math. Journal, 2024) recently discored new Garside structures. The PhD student will try using these Garside structures to (re)prove the faithfulness of the Khovanov-Seidel representation, following a proof strategy used in spherical type by Anthony Licata and Hoel Queffelec (Annales Scientifiques de l'ENS, 2021). Such faithfulness results imply in particular the faithfulness of the representation on Soergel bimodules, which is conjectural and has been much studied in the last twenty years.
Then, we will aim at generalizing the strategy when the number of atoms is infinite, to extend this strategy proof to a larger class of Artin-Tits groups. One of the possibilities is to extend the Artin-Tits group for it to contain a new element acting like a Garside element, which corresponds to the triangulated shift on the category on which the group acts.

Contexte de travail

The main goal is to prove the faithfulness of the Khovanov–Seidel representation in new cases, using Garside-type approaches. Developing extensions of Garside theory to the case of infinitely many atoms may also be of interest and will rely on the study of automorphism groups of certain triangulated categories.
Artin–Tits groups are classical objects in group theory and remain poorly understood. The categorical approaches we will use were introduced by Khovanov and Seidel in the early 2000s, inspired by symplectic geometry. This PhD project is part of a broader research program aiming to use categorical tools to derive geometric results about the autoequivalence groups of triangulated categories.

Contraintes et risques

No risk identified

Informations complémentaires

Profile and skills required
Master in pure mathematics, with familiarity with at least some of the following fields:
- braid and Artin-Tits groups;
- triangulated categories;
- homological algebra, categofication;
- geometric group theory.