University of Bristol, Wednesday May 20th, 2015
All talks will take place in Room SM2 in the School of Mathematics on University Walk. The School’s common room is located on the first floor.
After the last talk we will go for drinks and dinner.
If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk
The notion of complete reducibility was introduced by J.-P. Serre in 1998. It generalises the notion of a completely reducible module in classical representation theory. After giving an introduction to complete reducibility for algebraic groups, we discuss recent work with A. Litterick on classifying the subgroups of exceptional algebraic groups that are not completely reducible. The techniques used are a mix of standard representation theory, non-abelian cohomology and computational group theory.
Let Γ be a finite group. A natural and interesting way to think about the representation theory of Γ over an algebraically closed field k is to study the representation varieties Hom(Γ,GL(n,k)) for different n. These sets carry the structure of an affine variety, and the algebraic group GL(n,k) acts naturally. Understanding the orbit structure of this action allows one to use techniques from Geometric Invariant Theory to study representation-theoretic questions. For example, the closed orbits correspond to isomorphism classes of n-dimensional semisimple representations of Γ.
More generally, one can study the variety Hom(Γ, G) for any reductive group G. In this talk I’ll give a survey of some of the basic results in this area, including results of Richardson and Slodowy which allow one to relate the structure of Hom(Γ, G) to that of Hom(Γ, GL(n,k)), given an embedding of G in GL(n,k). Some of my own work has explored the limitations of Richardson’s techniques; in this vein I’ll also present an example which answers a question of Kulshammer.
A finite-dimensional representation of a group is said to be multiplicity-free if every irreducible representation appears at most once as a composition factor. I shall talk about the problem of classifying irreducible representations of simple algebraic groups for which the restriction to some proper subgroup is multiplicity-free. There are many interesting examples of such representations, and under suitable assumptions there is some hope of classifying them. There are also some nice connections with other aspects of algebraic group theory – for example, invariant theory and unipotent classes.