Algebraic Representation Theory

We are interested in the representation theory of algebras and groups arising in Lie Theory.

Examples of these are:

Algebraic and quantum groups
Weyl groups and Hecke algebras
Diagram algebras (eg the Temperley-Lieb and partition algebras)

The representation theory of such objects over a field of positive characteristic (or in the quantum case, when certain parameters are roots of unity) is not well-understood.

Diagram algebras typically arise in the context of statistical mechanices (see section to left), but have been shown to have strong connections with Lie Theory. One of our principal aims is to play off the sophisticated technical machinery developed in Lie Theory against the methods suggested by physics. More detailed information on the interplay between the two can be found here.

This work relies in large part on methods from the theory of finite dimensional algebras. In particular, many of the algebras of interest are either quasi-hereditary or cellular. In the former case, the study of "tilting modules" plays a central role. Determining the structure of such modules is in general difficult; however, in certain cases there are algorithms to do so.