On Picard groups of blocks of finite groups

TL;DR

This paper investigates the structure of Picard groups of blocks in finite groups, revealing local determination via fusion systems and describing their behavior over valuation rings, with applications to specific defect groups.

Contribution

It introduces a local description of Picard groups of blocks using fusion systems and characterizes their structure over valuation rings, extending understanding of block invariants.

Findings

Picard group subgroups are determined locally by fusion systems.

Picard groups over valuation rings are colimits of finite Picard groups.

Results applied to blocks with specific defect groups like cyclic or Klein four.

Abstract

We show that the subgroup of the Picard group of a $p$-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over the a complete discrete valuation ring ${\mathcal O}$ of characteristic zero with an algebraic closure $k$ of ${\mathbb F}_p$ as residue field is a colimit of finite Picard groups of blocks over $p$-adic subrings of ${\mathcal O}$. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.