移位Burnside双集函子的本质代数（带阿贝尔移位）
Essential algebra of the shifted Burnside biset functor with abelian shift
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中文总结 AI 辅助
本文研究带阿贝尔移位的移位Burnside双集函子的本质代数，给出分解准则并推广互素情形，证明其可分解为矩阵代数直和并参数化不可约模。
中文摘要 AI 辅助
设$T$为有限群，$kB_T$表示交换环$k$上的移位Burnside双集函子。我们研究$kB_T$在任意有限群$G$处的本质代数$\widehat{kB_T}(G)$。首先建立判定$G\times G\times T$的覆盖子群关于星积是否通过阶严格小于$|G|$的群进行分解的准则；第一个准则对$T$不作任何假设。作为应用，当$G$的任意非平凡商群不同构于$T$的某个截断时，我们完全描述了$\widehat{kB_T}(G)$，推广了Romero处理的互素情形。当$T$为阿贝尔群且$|T|$在$k$中可逆时，我们证明$\widehat{kB_T}(G)$分解为群代数$Out_T(A)$上的矩阵代数的直和，其中$A$遍历$G\times T$的约化子群的链类，并参数化了$\widehat{kB_T}(G)$的不可约模。
英文摘要
Let $T$ be a finite group and let $kB_T$ denote the shifted Burnside biset functor over a commutative ring $k$. We study the essential algebra $\widehat{kB_T}(G)$ of $kB_T$ at an arbitrary finite group $G$. We first establish criteria determining when a covering subgroup of $G\times G\times T$ factors, with respect to the star product, through a group of order strictly smaller than $|G|$; the first criterion places no hypothesis on $T$. As an application, we describe $\widehat{kB_T}(G)$ completely whenever no nontrivial quotient of $G$ is isomorphic to a section of $T$, extending the coprime case treated by Romero. When $T$ is abelian and $|T|$ is invertible in $k$, we prove that $\widehat{kB_T}(G)$ decomposes as a direct sum of matrix algebras over the group algebras of the groups $Out_T(A)$, where $A$ runs over the linkage classes of reduced subgroups of $G\times T$, and we parametrize the simple modules of $\widehat{kB_T}(G)$.
发表机构
- Boğaziçi University（博阿齐希大学）
- ADA University（阿达大学）
机构由 AI 辅助整理，请以论文原文为准。
