Overgroups of root groups in classical groups
I'll discuss work determining the overgroups of root groups in finite classical groups. This extends results of McLaughlin and Kantor, which describes the overgroups of long root groups under certain constraints. Much of the time will be devoted to motivation, including the place of this work in the description of the subgroup structure of classical groups and applications to the Palfy-Pudlak Question: is each finite lattice an interval in the subgroup lattice of some finite group?
Higher-rank invariants for finite group schemes
This is joint work with Eric Friedlander and Julia Pevtsova. We introduce higher rank variations on the notion of pi-points as defined by the second two authors for representations of finite group schemes. Using this we can define module of constant r-radical and r-socle type. Such modules determine bundles over the Grassmannian associated to the higher rank pi-points in the case that the group scheme is infinitesimal of height one. When the group scheme is an elementary abelian p-group, there is universal function for computing the kernel bundles as modules over the structure sheaf of the Grassmannian of r-planes in n space. These ideas also extend to elementary subalgebra of restricted p-Lie algebras.
Free resolutions of algebras
Any algebra has a canonical free resolution, the bar-cobar resolution. We construct a different resolution based on a presentation of the algebra as a quotient of a tensor algebra. This joint work with Alastair King is partly motivated by the desire to better understand Massey products in group cohomology.
On the representation dimension for group algebras
We will define the representation dimension of a finite-dimensional algebra and give some motivation. We give a global-local method to find bounds, for the case of group algebras and blocks. We apply this to blocks of symmetric groups with abelian defect groups; and as well to blocks with quaternion defect groups. Much of this is joint work with Petter Bergh.
Regularisation, homomorphisms and irreducible Specht modules
The question of which ordinary irreducible representations of the symmetric group remain irreducible in characteristic 2 was solved ten years ago by James and Mathas. Here we consider the generalisation to the Hecke algebra of type A, with q=-1. We present a conjectured solution, and describe work in progress aimed at a proof; this involves realising homomorphisms between Specht modules using the Khovanov-Lauda-Rouquier framework for Hecke algebras.
Soluble analogues of the Baer-Suzuki Theorem
We will discuss an approach to proving soluble analogues of the Baer-Suzuki Theorem without recourse to the Classification of Finite Simple Groups.
Principal W-algebras for GL(m|n)
(Joint with J. Brown and J. Brundan) W-algebras are certain associative algebras, whose representation theory has intimate connections with the representation of Lie algebras. The representation theory of W-algebras associated to reductive Lie algebras has attracted a great deal of recent research interest and major progress has been made. In this talk I will explain some first steps for W-algebras associated to reductive Lie superalgebras. Specifically, we will consider the case of the W-algebra associated to the principal even nilpotent element for GL(m|n), and consequences for the representation theory of the Lie superalgebra.
Variety isomorphism in group cohomology and control of p-fusion
I will report on recent joint work with Dave Benson and Ellen Henke where we show that if an inclusion of finite groups H < G of index prime to p induces an F-isomorphism of mod p cohomology rings, or equivalently a homeomorphism of cohomology varieties, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen in the p-nilpotent case, and also implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p = 2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p = 2.
Computing representations of finite groups and applications
We survey methods for computing character tables and representations of a finite group in characteristic 0 and over finite fields. An application of computing representations explicitly over number fields to the classification of maximal subgroups of classical groups over finite fields is described. Methods for the computation of the projective indecomposable representations of finite groups in finite characteristic, and applications to computing higher cohomology groups are discussed.
On 2-blocks with rank 2 abelian defect groups
We present a block theoretic analogue of a theorem of Richard Brauer on finite groups whose Sylow 2-subgroups are abelian of rank 2. This is joint work with C. Eaton, B. Kulshammer and B. Sambale.
Simple modules and quasi-hereditary twisted category algebras
The simple modules over a twisted category algebra admit a parametrisation in terms of Green relations and Schur functors, analogous to classical results in semigroup theory. The well-known classifications of simple modules over various diagram algebras, such as Brauer algebras, Temperley-Lieb algebras, and Jones algebras, can be obtained as special cases of this result. The notion of weights as in Alperin's weight conjecture extends to category algebras. Under some assumptions on the underlying category, the resulting weight algebras are quasi-hereditary. This is again a generalisation of a long list of results on diagram algebras. We conclude with a speculation on how to extend the group theoretic version of Alperin's weight conjecture to category algebras. This is joint work with Michal Stolorz.
Variations on the Ito-Michler Theorem
The Ito-Michler theorem is one of the fundamental results on the Character Degrees of Finite Groups. We will talk about several variations of it, related with field of values of characters, permutation characters or blocks.
Reduced fusion systems over 2-groups of sectional rank at most four
I will describe a result listing all reduced, indecomposable fusion systems over 2-groups of sectional rank at most four. This is of course motivated by the theorem of Gorenstein and Harada, where they listed all finite simple groups of sectional 2-rank at most four. The new result contains no surprises: the fusion systems in question are all those of simple groups on the Gorenstein-Harada list. But the method of proof seems very different: it is based on studying the different types of essential subgroups which can occur, rather than the centralizers of involutions. This also leads to a different way of organizing the final result, which I hope will be of interest.
Recognizing groups in characteristic 3
In this talk I will discuss recent joint work Gernot Stroth which identifies the isomorphism type of certain simple groups given partial information about the structure of the centralizer of an element of order 3. The talk will start by explaining why we require such identifications and then move on to discuss the identification methods used. The research discussed grows out of work developed by the Oxford group theory team in the late 1960s and early 70s.
Amalgamating fusion systems for different primes
Much of the focus of recent work on fusion systems been to focus on a single prime. In this talk we demonstrate that even by amalgamating two finite groups which individually have straightforward fusion systems for given primes (which are different), then we can realise an interesting group-theoretic configurations, which, although they can not occur in finite groups, are closely related to difficult configurations encountered in finite group theory. Furthermore, the (family of) groups we discuss each have many finite simple homomorphic images.
Clifford theory and algebraic group representations
Broadly speaking, Clifford theory is the study of the representation theory of a larger group G in terms of that of a normal subgroup N. Originally, the focus was on irreducible modules, but other modules, such is projective covers, may be studied this way, as in the well-known theorem of Alperin-Collins-Sibley. In algebraic group theory there is an extra impetus, since even simple algebraic groups can have infinitesimal normal subgroups which dramatically influence their representation theory. I will begin the algebraic groups study with the irreducible module case and give an old CPS Clifford theory proof of the Steinberg tensor product theorem. Then I will discuss the projective cover case, especially in light of recent progress on long-standing conjectures of Humphreys-Verma and Donkin.
Classical modules, simple modules and incidence matrices
Let G be a simple algebraic group in characteristic p > 0. By a classical G-module, we mean a dual Weyl module whose highest weight is a multiple of the first fundamental weight. If G = SLn then the classical modules are the symmetric powers of the standard module and the classical modules for other groups play a similar role. I will discuss recent work on the determination of the composition factors and submodule lattices of the classical modules for classical groups and for groups of type E6. These results have several applications. They yield the characters of all simple modules with highest weight a multiple of the first fundamental weight. Also, general formulae for the ranks of many incidence matrices arising from the geometry of finite classical groups are obtained. Some examples will be discussed.
A new "ZJ" theorem
Recently, George Glauberman has defined a new non-trivial characteristic subgroup, D*(S), of a finite p-group S. I will discuss our joint proof of the following analogue of the ZJ theorem:
Theorem: Let G be a p-stable group with F*(G) = O_p(G). If S is a Sylow p-subgroup of G, then D*(S) is a characteristic subgroup of G.
On the missing direction of Brauer's height zero conjecture
Brauer conjectured that all irreducible complex characters of a p-block b have height zero if and only if every defect group of b is abelian. The if direction of this conjecture is now proven by work of Berger-Knörr, and Kessar-Malle. In the talk I describe how the unproven direction can be reduced to a question on simple groups. This is joint work with G. Navarro.