Applications of Hecke Algebras: Representations, Knots and Physics – AHA

Hecke algebras originally appeared in the theory of modular forms in the 30's. Since then the name "Hecke algebras" has been progressively used to refer to a wide variety of objects, appearing and extensively studied in several areas of mathematics. Classes of examples of Hecke algebras of special interest for this project are:

- Centralisers (endomorphisms algebras) of induced representations;

- Deformations of Coxeter groups and (complex) reflection groups;

- Quotients of group algebras of (generalised) braid groups;

- Centralisers of tensor representations of quantum groups.

Quite remarkably, all the classes of examples above have a lot in common and this is the main reason why Hecke algebras are so important in modern mathematics: they can be studied from many points of view and they have applications in many different fields.

The project is centered on the study of Hecke algebras and concerns their applications to/interplay between different areas of
mathematics and physics. We will focus on three main areas, where Hecke algebras and related algebras play an important role:

- representation theory of different kind of Hecke algebras (and generalisations)

- low-dimensional topology, mainly the study of braid groups and invariants of links

- theoretical physics (integrable systems, statistical models, quantum field theory).

One major objective of this project is to study these three thematics simultaneously, especially focusing on interactions between them,
and considering Hecke algebras as bridges between these areas. We intend to bring together specialists of different themes related to Hecke algebras in order to benefit from their different approaches and points of view.
The project is multidisciplinary and interdisciplinary as we intend to obtain new results in each of the three areas, as well as develop and/or strengthen interactions between the areas, in particular between pure mathematics and theoretical physics

The proposal is divided into 6 research projects, two in each of the thematics of the proposal. These projects can be carried on independently, even though the main point of our proposal is that they are all related to each other to some extent. Thus interactions will naturally take place between the three thematics. The research projects are:

- Crystals for affine Hecke algebras and categorification

- Combinatorics for representations of Hecke algebras

- Yokonuma--Hecke algebras and links invariants

- Markov traces on quotients of braid groups

- Baxterization algebras

- Conformal field theory

The project was built in such a way that in each theme, at least as a starting point, some tasks rely on very recent works by the participants and for which we have already developed efficient methods. Then, our projects tend progressively to much more ambitious objectives, inside each thematic.
Our different projects will show connections with each others, through the fact that they are concerned with common structures, namely the Hecke algebras and related algebras, and through the fact that each thematic will bring different methods and points of view applicable to the others. Thus, interactions will naturally take place between the three thematics and the different tasks.