Geometry
Studying representations of infinite-dimensional objects is generally difficult. However, if they come with some additional structure, and we only consider representations respecting that structure, we can sometimes recover a manageable problem.
One possible choice is to impose some kind of analytic structure, and this is the strategy used for Lie groups. Over general fields this is no longer possible, and we instead appeal to algebraic geometry. Imposing the structure of a variety (or scheme) provides us with access to all of the sophisticated machinery developed by geometers.
Algebraic groups
An algebraic group is, roughly, a group which also has a (compatible) geometric structure. Thus one can apply methods from Algebra and from Geometry when attempting to study these objects. Although these can be studied over the complex numbers, the most interesting problems (at least among those which I am interested in) arise when we work with a field of positive characteristic. Typical examples of algebraic groups are matrix groups; for example the general linear or symplectic groups.
The study of representations of algebraic groups is very well established, although there are still many fundamental results which remain open. In particular we do not yet have much understanding of the simple modules; i.e. the irreducible components from which all other modules can be constructed. In general, there is not even a conjectural solution.
On the other hand, there is a natural class of modules which are very well understood - the Weyl modules. These are not simple in general, but they are known (in the sense that we can determine their characters), and an understanding of how they decompose into simple modules would be sufficient to determine all of the simples.
By the Steinberg tensor product theorem, it is enough (to solve the problem of determining the simple modules) to consider only Weyl modules labelled by 'small' weights. Consequently, the structure of general Weyl modules is a little less well understood. I have two papers considering this question; looking at the decomposition numbers of such modules and (jointly with Alison Parker) classifying the possible homomorphisms between them for the group GL(3).
Associated algebras
There are various algebras which can be associated to algebraic groups. In one direction we have quantum groups, which were originally introduced in Mathematical Physics. These are algebraic objects whose representation theory is very closely related to that of the corresponding algebraic groups, and which can be used as an auxilliary object of study. An example of an application of quantum groups in the classical setting cane be found in my work on distant decomposition numbers.
Another approach is to construct algebras directly from the algebraic group itself. Typically we look for finite dimensional algebras, which can then be studied both geometrically and combinatorially. The most important example of such an algebra is the Schur algebra, which links the representation theory of the general linear and symmetric groups.
By using the language of schemes, it is possible to produce new subgroups of algebraic groups which can be considered in the simpler setting of varieties. In particular, certain infinitesimal group schemes play an important role in the representation theory of algberaic groups. By combining this idea with the Schur algebra approach one arrives at the infinitesimal Schur algebras. I have some results on these algebras, but they remain very poorly understood.
Anton Cox
(A.G.Cox@city.ac.uk)
Last modified: Tue 7 Feb 2006