On simple modules in the Auslander-Reiten components of finite groups
type: Item
instance of: scholarly article
title: On simple modules in the Auslander-Reiten components of finite groups
zbMATH Open document ID: 0957.20003
published in: Mathematische Zeitschrift
publication date: 2001-03-20T00:00:00Z
review text: 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.
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 1488562
zbMATH Keywords: rank varieties
zbMATH Keywords: periodic modules
zbMATH Keywords: finite groups
zbMATH Keywords: wild blocks
zbMATH Keywords: group algebras
zbMATH Keywords: simple modules
zbMATH Keywords: stable Auslander-Reiten quivers
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: connected components
zbMATH Keywords: principal blocks
zbMATH Keywords: tubes
author: Shigeto Kawata
author: Gerhard O. Michler
author: Katsuhiro Uno
MaRDI profile type: Publication
full work available at URL: s002099900112
OpenAlex ID: W1975473773
DOI: 10.1007/S002099900112
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description: scientific article; zbMATH DE number 1488562