Representation theory over local rings

Lead Research Organisation: University of Manchester
Department Name: Mathematics

Abstract

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Publications

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Eaton C (2021) Donovan's conjecture and extensions by the centralizer of a defect group in Journal of Algebra

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Linckelman M (2021) Linear source invertible bimodules and Green correspondence in Journal of Pure and Applied Algebra

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Livesey M (2021) Arbitrarily large O -Morita Frobenius numbers in Journal of Algebra

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Livesey M (2022) Picard groups for blocks with normal defect groups and linear source bimodules in Journal of Algebra

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Livesey M (2021) On Picard groups of blocks with normal defect groups in Journal of Algebra

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Serwene P (2024) Proving a conjecture for fusion systems on a class of groups in Journal of Algebra

 
Description Picard groups for blocks of finite groups have recently become of greater interest to representation theorists partly because of new techniques that make their study more accessible, and because of the emergence of their importance in classification projects. Still little is known about them and basic questions remain to be answered. In work by Claudio Marchi (PhD student) and Michael Livesey, both together and separately, Picard groups have been determined in broad classes of groups (those with normal defect groups). Florian Eisele proved a key result that the Picard group (in the form used in representation theory) is finite, a result we had not expected to be able to prove.

Further, a question on Morita-Frobenius numbers was resolved by Michael Livesey and Florian Eisele, fundamental to our understanding of the diversity of types of blocks of finite groups. In particular they showed that these numbers can be arbitrarily large.
Exploitation Route Foundations have been laid in the study of Picard groups, which while resolving some questions has led to others that will be of interest for further research in the field of representation theory. For example, following Eisele's finiteness result, it is natural to ask for bounds on the size. Such questions are fundamental to our understanding of Morita equivalence. Further, an important open question on Morita-Frobenius number has been resolved, which immediately leads to more detailed questions that it would be interesting to answer with further research. For example, how quickly does the Morita-Frobenius number grow with the defect?
Sectors Creative Economy