On simple modules in the Auslander-Reiten components of finite groups
Let \(G\) be a finite group, let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(B\) be a wild block of the group algebra \(kG\). The paper under review is concerned with the position of simple modules in the stable Auslander-Reiten quiver \(\Gamma\) of \(B\). Two main results are proved. In the first, \(G\) is a perfect finite group of Lie type in characteristic \(p\). It is shown that each connected component of \(\Gamma\) contains at most one simple module. Moreover, the same is true for connected components isomorphic to \(\mathbb{Z} A_\infty\) without assuming that \(G\) is perfect. In the second main result, \(G\) is a finite group with an Abelian Sylow 2-subgroup, and \(B\) is the principal 2-block of \(G\). It is shown that any connected component \(\Theta\) of \(\Gamma\) contains at most one simple module \(S\), and that \(S\) lies at the end of \(\Theta\). Moreover, if \(\Theta\) is a tube then its rank can be computed explicitly.
scientific article; zbMATH DE number 1488562