Some examples of Picard groups of blocks

TL;DR

This paper computes specific Picard groups for certain 2-blocks with abelian defect groups, demonstrating their finiteness and triviality of Picent, and provides new general results on Picard groups of blocks.

Contribution

It presents the first calculations of Picard groups for these blocks and establishes new theoretical results on Picard groups with normal defect groups.

Findings

All such Picard groups are finite.

Picent group is trivial for these blocks.

Includes calculations for all abelian 2-groups of 2-rank at most three, except J1.

Abstract

We calculate examples of Picard groups for 2-blocks with abelian defect groups with respect to a complete discrete valuation ring. These include all blocks with abelian 2-groups of 2-rank at most three with the exception of the principal block of J1. In particular this shows directly that all such Picard groups are finite and Picent, the group of Morita auto-equivalences fixing the centre, is trivial. These are amongst the first calculations of this kind. Further we prove some general results concerning Picard groups of blocks with normal defect groups as well as some other cases.