线性源可逆双模与Green对应
Linear source invertible bimodules and Green correspondence
摘要：
我们证明了Green对应从有限群代数的块$B$的线性源Picard群$\mathcal{L}(B)$到线性源Picard群$\mathcal{L}(C)$诱导了一个内射群同态，其中$C$是$B$的Brauer对应.该同态将平凡源Picard群$\mathcal{T}(B)$映射到平凡源Picard群$\mathcal{T}(C)$。进一步证明了内置换源Picard群$\mathcal{E}(B)$关于$B$的亏群是有界的，并且当$B$有正规亏群$\mathcal{E}(B)=\mathcal{L}(B)$时.最后证明了任何可逆$B$ -双模的秩都是由$B$的秩有界的.
英文摘要：
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 $B$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
