标题： 有限群代数的一个块的源排列模块 显示英文标题

标题： The source permutation module of a block of a finite group algebra

摘要： 对于有限群$G$，域$k$为素数特征$p$，以及$S$为$G$的一个西罗$p$-子群，西罗置换模$\mathrm{Ind}^G_S(k)$在表示论和群论的多个方面起着作用，从 Alperin 的权猜想到$S$-$S$-双陪集在$G$中的统计考虑。 Sylow置换模在群代数$kG$的块分解上分解，但得到的块分量在 splendid Morita 等价下不是不变的。 我们引入了一个块分量的直和项，我们称之为源置换模，它被证明在这些等价下是不变的。 我们研究了源置换模的一般结构特性，并展示了将 Sylow 置换模的自内射性与 Alperin 权猜想相关的著名结果转移到源置换模上是可行的。 我们在各种情况下计算了这个模，例如某些具有循环缺陷群的块和具有 Klein 四元缺陷群的块，以及对称群的一些块，这是受到 Diaconis-Giannelli-Guralnick-Law-Navarro-Sambale-Spink 最近一篇关于对称群的 Sylow 置换模的自内射性问题的论文中一个提问的启发。 显示英文摘要

摘要： For $G$ a finite group, $k$ a field of prime characteristic $p$, and $S$ a Sylow $p$-subgroup of $G$, the Sylow permutation module $\mathrm{Ind}^G_S(k)$ plays a role in diverse facets of representation theory and group theory, ranging from Alperin's weight conjecture to statistical considerations of $S$-$S$-double cosets in $G$. The Sylow permutation module breaks up along the block decomposition of the group algebra $kG$, but the resulting block components are not invariant under splendid Morita equivalences. We introduce a summand of the block component, which we call source permutation module, which is shown to be invariant under such equivalences. We investigate general structural properties of the source permutation module and we show that well-known results relating the self-injectivity of the endomorphism algebra of the Sylow permutation to Alperin's weight conjecture carry over to the source permutation module. We calculate this module in various cases, such as certain blocks with cyclic defect group and blocks with a Klein four defect group, and for some blocks of symmetric groups, prompted by a question in a recent paper by Diaconis-Giannelli-Guralnick-Law-Navarro-Sambale-Spink on the self-injectivity of the endomorphism algebra of the Sylow permutation module for symmetric groups.