Representation theory of finite groups and connections to algebraic combinatorics

The representation theory of finite groups is a vast and active research area in algebra which has seen much exciting progress in recent years. This project will aim to tackle local-global questions in the representation theory of symmetric groups, their subgroups and other related groups. Symmetric groups are some of the most important and widely studied families of finite groups, occurring in all areas of science. They also serve as a fundamental example for which theories and conjectures for general finite groups are often first tested.

A lot of recent research in the representation theory of finite groups has focused on the Local-Global Conjectures, a family of open problems whose central theme is to understand how ‘global’ structures can be discerned from their ‘local’ or smaller parts. Even for groups as natural and ubiquitous as the symmetric groups, new inroads are still being made. For example, the McKay Conjecture, one of the cornerstones of this theory, only recently saw new proofs for symmetric groups that open the door to deeper, algebraic insights into local-global phenomena in group representation theory more broadly.

Another classical object that plays a core role in this area are the Sylow subgroups. Recent work into so-called ‘Sylow branching coefficients’, tools which lie on the interface between representation theory and algebraic combinatorics, seeks to shed light on local-global theory through a fresh lens. Indeed, symmetric groups are very closely connected to many topics in algebraic combinatorics, such as symmetric functions, and recent advances in the representation theory of symmetric groups have led to new progress on several long-standing problems in this area.

Applicants should have a strong background in group theory and representation theory, and should have an interest in algebraic topics with a combinatorial flavour.