Combinatorics
There are many beautiful combinatorial structures that arise in algebraic Lie theory. Indeed, there is a considerable amount of work developing algebraic constructions associated with various combinatorial structures such as symmetric functions.
However, the most common aspect of combinatorics that arises in my work is that related to the symmetric group. This is a very important object of study in its own right, but the combinatorics developed during its study has also arisen throughout the general area of diagram algebras, and elsewhere. One aspect of this has been formalised into the more general notion of a cellular algebra.
Symmetric groups
Understanding the representation theory of the symmetric groups is probably the most difficult problem in my area of research. The simple representations are still almost completely ill-understood. This is in sharp contrast to the analogous problem for algebraic groups where we have a conjecture when the characteristic is not too small, which is a theorem when the characteristic is large.
Despite the above remarks, some progress has been made. There are two main approaches to symmetric group representation theory. One is to consider the symmetric group in isolation, using explicit combinatorial constructions, while the other relates symmetric groups to general linear groups (or other algebraic objects).
My own work adopts the second point of view. This relates symmetric groups to certain quasi-hereditary algebras (the Schur algebras), and also generalises to quantum analogues of symmetric groups called Hecke algebras. In this way I was able to show how various decomposition numbers for the symmetric group can be related via an analogue of the tilting tensor product theorem for algebraic groups.
Anton Cox
(A.G.Cox@city.ac.uk)
Last modified: Mon 26 May 2008