# Representation Theory Group

The Representation Theory Group is interested in all aspects of modern representation theory. The focus is on gaining deep conceptual understanding of algebraic, combinatorial, geometric and topological structure.

## Membership

- Dr Anton Cox
- Professor Joseph Chuang
- Dr Maud De Visscher
- Professor Radha Kessar
- Professor Markus Linckelmann

## Postdoctoral researchers

- Dr Florian Eisele
- Dr Jorge Vitoria

## PhD students

- Samuel Creedon
- Niamh Farrell (completed 2016)
- Patrick Serwene
- Lleonard Rubio Y Degrassi (completed 2017)

## Expertise and research focus

The group has a broad range of expertise in mainstream modern representation theory.

The main areas of interest are: finite dimensional algebras symmetric groups and Hecke algebras, representations of finite and algebraic groups, Brauer and other diagram algebras, triangulated categories and dg categories, fusion systems, operads and homotopy algebras.

### Combinatorial and diagrammatic representation theory

Dr Cox and Dr De Visscher's research is concerned with relating two aspects of representation theory. On the one hand, the very classical theory for the symmetric group and algebraic groups; and on the other hand the more recently developed theory for diagram algebras.

On the classical side, Dr Cox and Dr De Visscher have studied the representation theory of algebraic groups, associated Schur algebras (and their quantized versions), Frobenius kernels, finite groups of Lie type and symmetric groups. On the diagram algebra side, in collaboration with Prof Martin (and Prof Doty), they have developed a comprehensive analysis of the representation theory of the Brauer (and walled Brauer) algebras. This theory has surprising links to the generalised Khovanov algebras and the study of Deligne's categories.

More recently, in joint work with Dr Bowman, Dr Enyang and Dr Orellana, Dr De Visscher has been investigating the famous Kronecker problem from classical representation theory, using a diagram algebra called the partition algebra. This brings a completely new approach to this longstanding open problem.

### Local-global representational theory

A major part of the research programme of Professors Chuang, Kessar and Linckelmann is concerned with understanding the local-global principles underlying the modular representation theory of finite groups. Roughly speaking, the local-global principal seeks to understand the relationship between the structural representation theoretic properties of a block of a finite group algebra over a field of positive characteristic p in terms of the structure and fusion patterns of the subgroups of the defect group of the block (i.e. the block fusion system).

There are many long standing and famous open conjectures in this context, such as the Alperin-McKay Conjecture, Braur's height zero conjecture, Donovan-Puig-Feit finiteness Conjectures, Alperin's weight conjecture and Broue's abelian defect group conjecture. Highlights of contributions to the global-local programme include the proof of Broue's abelian defect group conjecture for the symmetric groups by Chuang in joint work with Raphael Rouquier, the proof of the forward direction of Brauer's height zero conjecture by Kessar in joint work with Gunter Malle, and the proof of Puig's finiteness conjectures for cyclic blocks by Linckelmann. Much of the research of Professors Chuang, Kessar and Linckelmann is at the interface of representation theory with group theory, topology and geometry.

## Activities

The group maintains an extensive network of national and international collaborations (University of California Los Angeles, University of Chiba (Japan), National University Sinapore, University Hanover, University of Kaiserslautern, Ecole Polytechnique de Federale Lausanne, University of Valencia, York University, University of Leeds, University of Lancaster, Aberdeen University, University of Manchester). We are a member of the London Algebra Colloquium, a joint weekly seminar between City, Imperial College and Queen Mary University of London. In addition, along with our Phd Students and postdocs we run an informal Algebra reading group every term.