Representation theory over local rings

A group is an abstract structure which can arise in almost any area of mathematics or in physics. As such it is universal and can be a means of bridging disparate areas. Some examples of groups are the integers (with addition), the symmetries of a polyhedron (with composition of symmetries) or the fundamental group of paths on a surface. To understand these abstract objects, we need to represent a group in some way. We do this by considering it as a collection of transformations of space. The group may already have natural representations, as happens often in physics, e.g., orthogonal groups, or they may be obscure and involve transformations of very high dimensional spaces (for example the 'monster' sporadic group requires a 196,883 dimensional space). Further we need to study not just one representation of a group, but the entirety of the representations of that group. An object capturing this information is a module category.

Our interest is in the modular representations of a group, that is, those over a field of prime characteristic p. Here it makes sense to refine our module category. Instead of studying the group itself, we study its blocks. The study of the module category of a group amounts to study of the module category of each block in turn.

It has long been realised that rather than just study representations with respect to a field, it is beneficial to use a local ring as a bridge to connect representations in characteristic zero (classical representation theory) to those in characteristic p (modular representation theory). This approach has been so successful that we are increasingly studying representation theory with respect to local rings in its own right.

The overarching theme of this project is the exploitation of this approach in new ways, developing three interrelated bodies of theory aimed at shedding light on some of the big problems of modular representation theory.

One theory, which has been little explored, is to take certain quotients of blocks (i.e., smaller objects) which are just large enough to contain information that we are interested in with respect to whichever problem we are looking at. This can usually only be done in the context of local rings. A large part of this project will be laying the foundations of this approach, together with the calculations of examples needed to see patterns on which we can base theory. The famous Alperin-McKay conjecture from the 1970's is an example where this approach will be used.

Another theory is the study of the Picard group of a block, which is related to the block's self-similarities. The Picard group defined over a local ring is particularly amenable to study, as shown recently by Boltje, Kessar and Linckelmann, and has been used by Eaton to great effect to analyse module categories very precisely. A main theme of this project is to develop our understanding of Picard groups, and answer some outstanding question regarding their size and structure, as well as developing their application. The study of Picard groups of the quotient objects described above will further bring together the themes of the project.

The third theory concerns the realisation of modules and algebras of small fields and associated local rings and the relationships between them. This promises to be a powerful viewpoint for examining existing conjectures and Picard groups.

The main outcomes of the project will be on the one hand new theory and techniques which will spur further research, and on the other data about blocks, their Picard groups and their quotient objects, which will be incorporated into Eaton's website cataloguing blocks of finite groups.

The project involves knowledge of representation theory, group theory, homological algebra, and number theory, and will benefit from collaborations with the strong algebra community both in the UK and outside.

The impact within representation theory will be twofold. First, recent momentum in the development of the understanding of Picard groups and structure of blocks, and on the role of rationality arguments, will be maintained. This momentum will be further maintained through the training of two PDRAs, collaboration and through the workshop. Secondly a new area, that of quotient objects defined using character idempotents, will be introduced, with first examples and fundamental results. The proposal offers fresh perspectives on long standing open questions of representation theory around which there is currently a great deal of activity. These new perspectives will shape the future direction of the subject.

The proposal includes a workshop to be held in 2021, midway through in the project. By this stage the PDRAs will be established and avenues of research cemented. It will bring together potential collaborators beyond those already identified and create a focus on the set of problems and approaches which form this project. Collaboration will be an important part of the project, with the attendant knowledge transfer and exchange of ideas.

Long term impact is hard to predict, but one pathway is as follows. In the medium term the proposed research will feed into the large effort to understand global-local determination in blocks. A global-local theory would have potential for wide applications because it allows the reduction of problems to simple local situations. Such situations could arise whenever symmetry is involved.