Linear source invertible bimodules and Green correspondence

TL;DR

This paper explores how the Green correspondence induces an injective homomorphism between linear source Picard groups of blocks in finite group algebras, revealing structural bounds and relations with defect groups.

Contribution

It establishes an injective homomorphism induced by the Green correspondence between linear source Picard groups and analyzes bounds related to defect groups.

Findings

Injective homomorphism from al{L}(B) to al{L}(C) via Green correspondence

Bound on endopermutation source Picard group al{E}(B) in terms of defect groups

Bound on the rank of invertible B-bimodules

Abstract

We show that the Green correspondence induces an injective group homomorphism from the linear source Picard group $\mathcal{L}(B)$ of a block $B$ of a finite group algebra to the linear source Picard group $\mathcal{L}(C)$, where $C$ is the Brauer correspondent of $B$. This homomorphism maps the trivial source Picard group $\mathcal{T}(B)$ to the trivial source Picard group $\mathcal{T}(C)$. We show further that the endopermutation source Picard group $\mathcal{E}(B)$ is bounded in terms of the defect groups of $B$ and that when $B$ has a normal defect group $\mathcal{E}(B)=\mathcal{L}(B)$. Finally we prove that the rank of any invertible $B$-bimodule is bounded by that of $B$.