Group representations: orders, lattices and deformations

Groups are algebraic structures that encode symmetries of mathematical objects. These objects may be very concrete geometric shapes like polyhedra, abstract algebraic structures like rings or fields, or even physical systems. Groups are therefore of fundamental importance in many areas of mathematics and physics. Some groups are naturally represented as symmetries of a space. For instance, the symmetry group of a polyhedron consists of rotations and reflections of the Euclidean space the polyhedron lives in. But many other groups are not of this form, and the main thrust of representation theory is to describe how a given abstract group can be represented as symmetries of some vector space in terms of linear maps or matrices. This project focuses on the representation theory of finite groups on vector spaces over fields of positive characteristic, or lattices over rings closely related to the integers. This area is called the “modular representation theory of finite groups”, and it is one of the major research directions within representation theory due to the rich and difficult structure of such representations and multiple strong and well-evidenced conjectures, including Donovan’s conjecture which we will address in this project. Using the terminology of the field, the overarching aim is to understand “block algebras” and their “module categories”, which encode their representations. This project is subdivided into three work packages. In the first package we will broaden the scope and look at the modular representation theory of profinite groups – a class of topological groups closely related to finite groups. Profinite groups appear naturally in number theory, but their modular representation theory is still being developed. Moreover, there are striking structural similarities between block algebras of finite groups over p-local rings and block algebras of profinite groups. I will explore these structural similarities and develop the theory of blocks of profinite groups further. There is evidence that Donovan’s conjecture may have a much cleaner answer in the profinite setting, and exploring this idea is a main aim of this project. Profinite groups are ubiquitous across mathematics, in particular in number theory, so results on their modular representation theory will have fruitful applications elsewhere. In the second work package we will study the deformation theory of block algebras defined over a p-local ring. Deformation theory is a powerful tool in the representation theory of finite-dimensional algebras, and my own research has generalised that to the mixed-characteristic setting encountered in block algebras defined over a p-local ring. This approach has already had an impact on Donovan’s conjecture, and in this project we will develop this approach further with a view towards not just Donovan’s conjecture but also the Donald-Flanigan conjecture – the main conjecture on the deformation theory of block algebras. Both work packages above are linked to the representation theory of finite-dimensional algebras in terms of methodology and approach. The proposed work will reexamine the links between the representation theory of finite groups and finite-dimensional algebras, two areas which have diverged in the past few decades. In the third work package we will apply all of the above to make headway on a general reduction for Donovan’s conjecture in the classical finite group setting. This is very ambitious, but good progress has been made in recent years and results will have a major impact on the whole field.