Picard groups for blocks

Abstract

This thesis concerns the study of Picard groups for blocks, in the context of modular representation theory of finite groups. The first chapter establishes the notation and provides an introduction to modular representation theory of finite groups, with a special focus on Morita equivalences. In Chapter 2 we address the problem of calculating Picard groups. The results of this chapter rely on the existence of stable equivalence of Morita type and on methods that were successfully used by other authors to provide the first examples of Picard groups for blocks. In Chapter 3 we investigate Picent for blocks, proving that it is trivial for a perhaps surprisingly large family of blocks. We also provide examples of blocks with non-trivial Picent, even with normal abelian defect group and abelian inertial quotient. The content of this chapter is joint work with Michael Livesey. In Chapter 4 we prove that, for blocks with normal defect groups in odd characteristic, any bimodule inducing a Morita auto-equivalence must have endopermutation source, providing evidence to an existing open problem. We also have partial results for blocks with normal defect groups in even characteristic. The content of this chapter is joint work with Michael Livesey.

Date of Award	1 Aug 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Charles Eaton (Supervisor) & Michael Livesey (Supervisor)