4th meeting: Algebraic Groups

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

2:00-3:00 Adam Thomas (Cambridge) Complete reducibility in exceptional algebraic groups

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.

3:00-3:30 Break

3:30-4:30 Michael Bate (York) Representation varieties, reductive pairs and a question of Kulshammer

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.

4:30-5:30 Martin Liebeck (Imperial) Multiplicity-free representations of algebraic groups

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.

2nd meeting: Representations of Groups

University of Birmingham, Wednesday January 14th, 2015

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

2:00-3:00 David Stewart (Cambridge) Cohomology for finite and algebraic groups

The representation theory for finite groups of Lie type in defining characteristic is rather complicated, but owing to the link with algebraic groups, there are quite a few tools. I want to give an overview of some of the theory and show how some of these can be used to bound the cohomology of finite groups with coefficients in simple modules, motivated by an old conjecture of Guralnick.

3:00-4:00 Melanie de Boeck (Kent) Foulkes modules for the symmetric group

The action of the symmetric group \mathfrak{S}_{mn} on set partitions of sets of size mn into n sets of size m gives rise to a permutation module called the Foulkes module.  Structurally, very little is known about Foulkes modules, even over \mathbb{C}.  In this talk, we will introduce Foulkes modules and their twisted analogues before presenting some results which shed light on the irreducible constituents of the ordinary characters of twisted Foulkes modules.

4:00-4:30 Coffee

4:30-5:30 Gunter Malle (Kaiserslautern) Local-global conjectures in the representation theory of finite groups

More than 60 years ago Richard Brauer developed the theory of representations of finite groups over arbitrary fields. It showed a strong connection between the representation theory of a finite group and that of its p-local subgroups, for p a prime. Many more such connections have been observed in the meantime, but most of these are still conjectural.

Recently, a new reduction approach has offered the hope to solve all of these fundamental conjectures by using the classification of finite simple groups.  In our talk we will try and explain the nature of these problems and will report on recent progress which might eventually lead to a solution of these long standing fundamental questions.

1st meeting: Groups and Geometry

Birkbeck, University of London, Wednesday November 5th, 2014

Room 745, Malet Street

If you wish to attend, please contact b.fairbairn [at] bbk.ac.uk

1:00-2:00 Mairi Walker (Open University) Strong approximation in Fuchsian groups: A geometric approach

A rational number is a strong approximation for an irrational number if there is no closer rational number with smaller denominator. It is well known that the strong approximants of an irrational number are given by the convergents of its simple continued fraction expansion. In this talk we look at how the idea of strong approximation can be generalised to the approximation of irrational numbers by fixed points of parabolic elements of Fuchsian groups, taking a geometric approach. In particular, we will show that strong approximants by parabolic fixed points of the Theta group are precisely the convergents of even-integer continued fractions.

2:00-3:00 Goran Malic (Manchester) Partial duality of maps on surfaces and their monodromy and automorphism groups

Recently S. Chmutov introduced an operation on maps on surfaces called partial duality which generalises the usual notion of map duality. Partial duals preserve connectedness and the number of edges, but not necessarily the number of vertices and faces, and therefore the genus of the embedding is not necessarily preserved. Using the combinatorial model for maps on surfaces we will look at how partial duality affects the monodromy and automorphism groups of the original map. For example, we can trivially observe that a map will have abelian monodromy if and only if its partial duals have abelian monodromy.

3:00-3:30 Coffee

3:30-4:30 Bill Harvey (King’s College London) Automorphisms of hyperbolic surface bundles

The recent landmark results of Kahn and Markovic have completed an epic voyage (steered largely by the amazing insights of Bill Thurston) through geometric topology in dimensions 2 and 3, and we now know in principle everything about the structure of compact hyperbolic 3-manifolds: they are virtual surface bundles with pseudo-Anosov monodromy. This talk will survey some of the background and try to answer the obvious question: what (necessarily finite) automorphism groups can occur for these 3-manifolds?