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Details of Grant 

EPSRC Reference: EP/W028794/1
Title: Exotic Representation Theory
Principal Investigator: Semeraro, Dr J
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics and Actuarial Sciences
Organisation: University of Leicester
Scheme: New Investigator Award
Starts: 01 October 2022 Ends: 30 September 2025 Value (£): 356,861
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
08 Feb 2022 EPSRC Mathematical Sciences Prioritisation Panel February 2022 Announced
Summary on Grant Application Form
This proposal lies at the intersection of Geometry, Topology and Algebra. Through the study of certain generalised finite groups which arise naturally in each of these contexts, I will forge new interdisciplinary connections between them. These connections will serve to sustain and develop links between algebra and related disciplines, enhancing the key underpinning role that algebraic, geometric and topological research plays across the mathematical sciences, as well as the in UK's world-leading position in the area. They will also and offer potential applications to Quantum Mechanics, Topological Data Analysis and Artificial Intelligence.

The study of groups begins with the study of symmetries of shapes such as regular polygons or 3-dimensional polyhedra like the cube, tetrahedron or icosahedron. As for these examples, many interesting symmetry groups arise as reflection groups - those generated by reflections (linear transformations fixing a hyperplane of the underlying vector space). A reflection group is called `real' or `complex' according to whether the vector space is real or complex. Both the real and irreducible complex reflection groups have been classified.

The classification of finite simple groups, the building blocks of symmetry, reveals that `most' simple groups are of Lie type, meaning that they are, in some sense, determined by their Weyl groups which are real reflection groups. For example, the groups of rotations and reflections of higher dimensional tetrahedra (the symmetric groups) give rise to simple groups of matrices defined over some finite field.

This project will study the representation theory of spetses, which are yet undefined objects generalising the simple groups of Lie type in which the associated Weyl group is assumed to be a complex reflection group. Despite the absence of an actual group with which to perform calculations, techniques from Algebra and Topology - specifically the theories of Hecke algebras and $p$-compact groups - will be combined to garner new representation-theoretic information (such as character degrees, decomposition matrices and Brauer trees).

I will show that this information is rich enough to satisfy generalised local-global counting conjectures from the modular representation theory of finite groups. For groups, these conjectures will follow as special cases - when the Weyl group is real - thereby settling questions that have been around for more than half a century.

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Organisation Website: http://www.le.ac.uk