Local-global conjectures for spetses (Ref: MA/JS-SF1/2026)

About the Project

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 which 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 something more general, now associated to a complex reflection group, to which similar data can be assigned. The theory of spetses date 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 this project.

Name of primary supervisor/CDT lead:
Jason Semeraro j.p.semeraro@lboro.ac.uk

Entry requirements:
Students should have, or expect to achieve, a 2:1 in mathematics or a closely related discipline OR any upper-second class (2:1) honours degree and a Master’s degree at merit in a relevant discipline.

English language requirements:
Applicants must meet the minimum English language requirements. Further details are available on the International website (http://www.lboro.ac.uk/international/applicants/english/).

Bench fees required: No

Closing date of advert: 31st July 2026

Start date: October 2026

Full-time/part-time availability: Full-time 3 years

Fee band: 2025/26 Band RA (UK £5,006, International £22,360)

How to apply:
All applications should be made online. Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MA/JS-SF1/2026 in your application. To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents.
The following selection criteria will be used by academic schools to help them make a decision on your application. Please note that this criteria is used for both funded and self-funded projects.
Please note, applications for this project are considered on an ongoing basis once submitted and the project may be withdrawn prior to the application deadline, if a suitable candidate is chosen for the project

Project search terms:
pure mathematics, representation theory, group theory, topology

Email Address Sci:
sci-pgr@lboro.ac.uk