Programme

Algebraic Groups

In this three-hour long series of talks we will explain the basic concepts behind algebraic 𝕜-groups, where 𝕜 is a field, and survey the basic structural results. The main focus will be on the case where 𝕜 is a finite field.

Exercises

Characters and blocks of finite groups

This course provides an introduction to the ordinary and modular representation theory of finite groups through the study of characters and Brauer p-blocks. We will start with the fundamentals of character theory, emphasizing the role of characters as powerful tools to study finite groups. The second part is devoted to Clifford theory, which explains how characters behave under restriction to normal subgroups. We will present important results as Clifford and Gallagher's theorems in this part. In the final part, we will move into modular representation theory, introducing Brauer characters, decomposition matrices, and Brauer p-blocks. Throughout the course, the aim is to combine concrete examples with the general theory. 

Exercises

Characters of Groups of Lie Type

In this series of three hour-long talks, we’ll discuss some of the ways that the characters of groups of Lie type can be described and parametrized. In the third hour, we will also discuss some of the relationships between these and the $p$-blocks for a prime $p$ distinct from the defining characteristic of the finite group of Lie type.

Exercises

Brauer's height zero conjecture

The talk will try to explain some ingredients in the recently completed proof of Brauer's height zero conjecture, building upon notions, concepts and results presented in the mini-courses.

On an extended use of the Green correspondence to calculate Brauer trees

A fundamental problem in the representation theory of finite groups is to construct all irreducible representations (resp. (Brauer) characters) of a given finite group. Determining their degrees is the first step. Over a field of positive characteristic p this problem is an open problem and it is challenging even for families of groups such as the symmetric groups.

Now, to address this question, it is standard to consider the $p$-modular decomposition matrix of the (p-blocks of the) group, which arises from writing the ordinary irreducible characters in terms of the irreducible p-Brauer characters. For a $p$-block with cyclic defect groups this information can be read off from its "embedded Brauer tree": a planar embedded graph encoding its Morita equivalence class.

For the sporadic simple groups and their covering groups most trees were calculated by Hiss and Lux (1997). However, some trees are incomplete and the problem of finishing off their calculation is still open. Here, it should be noted that most trees were automatically derived from computer algebra methods. When these methods were not powerful enough, further theoretical arguments, such as a clever use of the Green correspondence, was used. More precisely, it was possible to obtain information about the shape of the tree using the Green correspondents (at the block level) of the simple modules (and their Heller-translates) over the normalisers of a defect group.

Thanks to results of Bleher and Chinburg (2000) on the Auslander-Reiten quiver of Brauer-tree algebras, it is in fact possible to calculate explicitly the Green correspondent of any indecomposable module of a p-block with cyclic defect groups, with pen and paper. The aim of this talk is to show how this method can be used as an alternative to condensation methods, for example, to finish off the calculation of the Brauer tree of the principal 19-block of the Thompson group.

Local-global conjectures and fusion systems

Let p be a prime and G be a finite group. There is a pervasive global-local phenomenon in which representation-theoretic information about G is captured by that of its p-local subgroups (G-normalisers of non-trivial p-subgroups). The local side of conjectures which make this precise can oftentimes be rephrased using the p-fusion category F of G, which is a rich topological invariant. I will explain how this works for the Alperin Weight Conjecture, and then exhibit a completely topological approach to the determination of F when G is a finite reductive group. If time permits, I will discuss an extension of this approach to fusion systems of unipotent p-blocks of such groups.

Block enumeration

In modular representation theory, some central questions naturally reduce to the study of blocks. The complexity of a block is controlled by a statistic called the defect.

In this talk, we are interested in counting the simplest (i.e. defect 0) blocks of certain finite groups and related Hecke algebras.

I will recall some classical results (based on the combinatorics of core partitions) and present some recent developments.

This is a current project involving Emily Norton, Nathan Chapelier, Nicolas Jacon, Cédric Lecouvey and Sylvie Corteel.

Complete reducibility and subgroup structure of 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 a thorough introduction to complete reducibility for algebraic groups, we discuss its impact on questions about the subgroup structure of algebraic groups. This includes recent and ongoing work with A. Litterick and R. Pengelly 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.

On the Lie algebra structure of derivations on finite group algebras

Any associative algebra over a commutative ring gives rise to a Lie algebra obtained by considering derivations on the algebra. This talk is about exploring the connections between the associative algebra structure of finite group algebras and the corresponding Lie algebras of derivations. The main motivation for pursuing these connections in finite group representation theory is the fact that these Lie algebras have very good invariance properties that may help with determining some invariants of the associative algebra structure, such as the number of simple modules. We present some of the broader context, calculations for specific classes of finite group algebras, and some conjectures.

Partial character table for Z_ℓ-spetses

Given a prime ℓ, an ℓ-adic reflection group W of order prime to ℓ, and a prime power q congruent to 1 modulo ℓ, we construct a table of values which in the case that W is the Weyl group of a finite reductive group 2G(q) specialises to the piece of the character table of G(q) where the rows correspond to characters in the principal block and the columns correspond to conjugacy classes of ℓ-elements. We conjecture and in some cases prove that our character values satisfy orthogonality relations consistent with the character theory of finite groups. If we restrict to rows corresponding to unipotent characters, we can drop the assumptions on q and ℓ. This is joint work with Gunter Malle and Jason Semeraro, and is part of a larger project enriching the Broué-Malle-Michel theory of spetses via the theory of fusion systems.