Local-Global Conjectures for Spetses

A real reflection group is a finite group of matrices with real entries generated by reflections. Such groups occur ubiquitously as Weyl groups in the classifications of simple complex Lie groups and algebraic groups, and in the theories of braid groups, knots, and Hecke algebras. A complex reflection group is a finite group of matrices with complex entries generated by pseudo-reflections - matrices that fix a hyperplane. Certain data in the representation theory of a rational reductive group, for example its unipotent representation degrees, can be given generically in terms of the Weyl group. A “spets” is a more general object, associated with a complex reflection group, to which similar data can be assigned. The theory of spetses dates back to the 1990s, but exciting recent developments in algebraic topology have broadened its remit to include generalised theories of unipotent blocks, fusion systems, and characters. A wealth of questions then present themselves, most pertinently whether the local-global conjectures - a series of far-reaching and important open problems in modular representation theory - can be extended and proved in this generalised setting. This will be a key focus of the project. 94% of Loughborough’s research impact is rated world-leading or internationally excellent. REF 2021

The standard University IELTS English language requirement is 6.5 overall with 6.0 in each individual element (reading, writing, listening and speaking).

International students: £23,100

UK students: £5,238

UK/Home: £5,238

International: £23,100

October 2026

31 July 2026

• Loughborough, United Kingdom