# Research topics

## In 'Algebra'

Back to 'Mathematics Research'The Algebra Group has broad interests in a number of areas, mostly connected to group theory and Lie theory.

This page collects a few of the areas that the Group specialises in, with some information about research conducted in those areas.

## Fusion systems

David Craven has studied fusion systems, from both a structural point of view and looking at specific classification problems. The most important result from the structural point of view is the theorem that a weakly normal subsystem always contains a normal subsystem on the same subgroup. This implies that two of the three possible notions of a simple fusion system coincided, and simplifies a lot of computations, as it can be tricky to prove that a weakly normal subsystem is normal.

More recently, in joint work with Bob Oliver and Jason Semeraro, he classified all simple (saturated) fusion systems on *p*-groups with an elementary abelian subgroup of index *p*. This classification produced many new examples of exotic fusion system, proved that there are unboundedly many exotic simple fusion systems on the same *p*-group, and found new examples of simple fusion systems that nevertheless have non-trivial quotient systems.

This classification intersects collections of saturated fusion systems discovered by Chris Parker and his PhD student Murray Clelland. Chris is motivated to study fusion systems by his interest in *p*-local structures in finite simple groups. His expertise in this area inspired the construction with Gernot Stroth of a family of exotic systems with an extraspecial subgroup of index *p*. Chris's student Raul Moragues Moncho went on to determine all the simple fusion systems on such *p*-groups.

More recently, with Valentina Grazian, Chris has produced a classification of saturated fusion systems defined on maximal class *p*-groups. In a slightly different direction, in joint work with Jason Semeraro, Chris has developed computational methods to work with fusion systems using computers.

## Modular representation theory of groups

David Craven has worked on the representation theory of finite groups over fields of positive characteristic for most of his career. He started with work on understanding summands of tensor products of representations of groups, and then soon after moved on to Broué's abelian defect group conjecture.

A big part of this work is on perverse equivalences. Arising out of work of Chuang and Rouquier that solved Broué's conjecture for symmetric groups, perverse equivalences are a combinatorially defined derived equivalence. The aim is to give an explicit potential derived equivalence in the case of groups of Lie type: in joint work with Raphaël Rouquier of UCLA, the first properties of these equivalences were computed. This breakthrough has spurred a significant amount of new research into the topic, and ten years later we have much more detailed knowledge about the decomposition numbers of unipotent blocks than could have been imagined before.

More recently, he has applied modular representation theory to the difficult question of understanding the subgroup structure of the exceptional groups of Lie type.

In 2019 he wrote *Representation Theory of Finite Groups: a Guidebook*, an attempt to survey the modern theory of representations of groups in less than 300 pages, up to the very latest in research.

## Representations of algebraic groups and Lie algebras

Groups of continuous transformations, now known as Lie groups, have their origins in the work of Sophus Lie in the late 19th century, partly motivated by the wish to develop Galois theory for differential equations. Since then, the study of Lie groups and Lie algebras has been a central theme in mathematics. In the 1950s, the ``analytic theory’’ was extended so that it also makes sense over arbitrary algebraically closed fields and spawned the area of mathematics now known as algebraic Lie theory. Nowadays, algebraic Lie theory encompasses a variety of different areas; for example, it includes the structure theory and representation theory of algebraic groups, finite groups of Lie type, Lie algebras, and many related algebras. A common theme in the area is a strong interplay between algebra, geometry and combinatorics.

Throughout his career, Simon Goodwin has developed interests in a range of topics in Lie theory and representation theory. His research is centered around questions in the representation theory of Lie algebras and algebraic groups. Recently he has been interested in the representation theory of modular Lie algebras and of Lie superalgebras, and in particular the application of the theory of W-algebras to these areas. This research has involved a blend of algebraic, combinatorial and geometric methods.

When studying the representation theory of semisimple algebraic groups and Lie algebras over fields of positive characteristic, one difference from the theory in characteristic zero is that there now exist finite-dimensional simple modules for the Lie algebra which aren't seen by the algebraic group. We call these representations non-restricted. Matthew Westaway’s research has largely focused on deepening our understanding of these simple modules. Among his contributions, he has developed a theory of non-restricted representations which is compatible with higher Frobenius kernels (mirroring the known compatibility with the first Frobenius kernel) and has generalised some key aspects of the theory of restricted representations to the non-restricted world. Since the representation theory of finite W-algebras provides a characteristic zero analogue of the non-restricted representation theory of Lie algebras, he also works in this area, in which he collaborates with Simon Goodwin and Lewis Topley (Bath).

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