Picard groups for some blocks with TI defect groups

TL;DR

This paper computes the Picard groups for certain blocks with trivial intersection defect groups and cyclic inertial quotient, using generalized methods and Green correspondence to support a conjecture on auto-Morita equivalences.

Contribution

It extends the calculation of Picard groups to blocks with TI defect groups and cyclic inertial quotient, providing evidence for a conjecture on auto-Morita equivalences.

Findings

Picard group equals the group of auto-equivalences, (B)

Methods generalize previous results on self-stable equivalences

Supports conjecture relating Picard groups and auto-Morita equivalences

Abstract

We calculate the Picard groups for principal blocks $B$ with TI defect groups and cyclic inertial quotient. The methods used generalize results on self stable equivalences and take advantage of the existence of equivalences given by Green correspondence in this setting. In particular, we show that $\text{Pic}(B)=\mathcal{E}(B)$, giving more evidence for a conjecture on basic auto-Morita equivalences.