Endo-permutation modules as sources of simple modules.
Endo-permutation modules as sources of simple modules.
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内排列模块作为简单模块的来源。
DOI：
10.1515/jgth.2003.033
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发表时间：
2003
影响因子：
0.5
通讯作者：
N. Mazza
中科院分区：
文献类型：
--
作者：
N. Mazza
对于有限$p$-可解群$G$和素数特征为$p$的代数闭域$k$，单$kg$-模的源是内置换模(见{Pu1}或{Th})。L.Puig证明了这个源一定同构于形式为$\BigoTimes_{Q/R\in\cal S}\10^P_Q\inf^Q_{Q/R}(M_{Q/R})$的内置换模的帽，其中$M_{Q/R}$是具有顶点$Q/R$的不可分解扭转内平凡模，$\cal S$是单$kg$-模的顶点的循环、四元数和半二面体截面的集合.目前，人们猜想，如果一个单模的源是内置换模，那么它应该具有这样的形状。在这篇文章中，我们将给出一个方法，它允许我们显式地实现任何这样的不可分解模的上限作为有限$p$-幂零群的单模的源。
The source of a simple $kG$-module, for a finite $p$-solvable group $G$ and an algebraically closed field $k$ of prime characteristic $p$, is an endo-permutation module (see~\cite{Pu1} or~\cite{Th}). L. Puig has proved, more precisely, that this source must be isomorphic to the cap of an endo-permutation module of the form $\bigotimes_{Q/R\in\cal S}\Ten^P_Q\Inf^Q_{Q/R}(M_{Q/R})$, where $M_{Q/R}$ is an indecomposable torsion endo-trivial module with vertex $Q/R$, and $\cal S$ is a set of cyclic, quaternion and semi-dihedral sections of the vertex of the simple $kG$-module. At present, it is conjectured that, if the source of a simple module is an endo-permutation module, then it should have this shape. In this paper, we are going to give a method that allow us to realize explicitly the cap of any such indecomposable module as the source of a simple module for a finite $p$-nilpotent group.