Linear source invertible bimodules and Green correspondence
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 $B$.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.10131
 Bibcode:
 2020arXiv200410131L
 Keywords:

 Mathematics  Representation Theory;
 20C20 (Primary) 16D90 (Secondary)