Picard groups for blocks with normal defect groups and linear source bimodules

TL;DR

This paper proves that bimodules inducing Morita auto-equivalences of blocks with normal defect groups have linear sources, advancing understanding of their structure in various characteristics.

Contribution

It establishes that such bimodules always have linear sources for blocks with normal defect groups in odd characteristic and under certain conditions in characteristic 2.

Findings

Bimodules inducing Morita auto-equivalences have linear sources in odd characteristic.

The result extends to characteristic 2 for specific defect group structures.

Provides new insights into the structure of blocks with normal defect groups.

Abstract

It is an open problem as to whether any bimodule inducing a Morita auto-equivalence of a block must have endopermutation source. We prove that, for blocks $b$ with normal defect groups in odd characteristic, a stronger result holds, namely that all such bimodules have linear source. We also prove the analogous result in characteristic $2$, provided that the defect group is of a specific, slightly restrictive, form.