Hecke algebras and categorical representation theory

The first main goal of the project is a categorical presentation of the singular Hecke algebras by diagrammatic generators and relations. In order to prove such a presentation, we also plan to construct a cellular-type basis.

The presentation will then be applied to rational representation theory of reductive groups in positive characteristic, especially to compute characters and cohomological dimensions.

The project also aims to develop categorical methods in representation theory (or 2-representation theory), such as internal hom techniques, and obtain classification results for simple 2-representations.

At the same time, we plan to explore the applications of categorical methods to representation theory of reductive groups, quantum groups, and Lie algebras.

Some specific problems in the project are Kostant´s problem for simple modules over Lie algebras and representation stability of linear algebraic groups via polynomial functors.

The project utilizes established international collaborations, drawing on our collaborators´ expertise to achieve our goals.

Our methods include category theory, homological algebra, algebraic combinatorics, and computations using computer algebra systems.

The main results of the project should have applications in algebraic geometry, low-dimensional topology and theoretical physics, as well as in representation theory.

It is also expected that side results obtained in the project have applications in algebra, category theory, and combinatorics.