The Auslander-Reiten quiver of a modular group algebra revisited
type: Item
instance of: scholarly article
title: The Auslander-Reiten quiver of a modular group algebra revisited
zbMATH Open document ID: 0693.20008
DOI: 10.1007/BF02571322
author: Christine Bessenrodt
published in: Mathematische Zeitschrift
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174204
review text: Let G be a finite group, F a field of characteristic \(p>0\). In an important paper [Math. Z. 179, 97-121 (1982; Zbl 0479.20008)] \textit{P. Webb} proved that the tree class of a connected component of the stable Auslander-Reiten quiver of the group algebra FG is either a Dynkin diagram \(A_ n\), a Euclidean diagram or one of the infinite trees \(A_{\infty}\), \(B_{\infty}\), \(C_{\infty}\), \(D_{\infty}\), \(A^{\infty}_{\infty}\). For \(p\neq 2\), \textit{T. Okuyama} [J. Algebra 110, 425-430 (1987; Zbl 0624.20006)] proved that the tree class is never a Euclidean diagram. In this paper it is shown that for \(p=2\) only the Euclidean diagrams \(\tilde A_{12}\) or \(\tilde B_ 3\) are possible; it is known that these diagrams occur for \(G={\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2\) resp. \(G=A_ 4\) and \(F=GF(2)\). More precisely, it is proved that the block containing such a Euclidean component has a Klein four group as a defect group. For the reduced graph of a component also the Euclidean type \(\tilde A_ n\) is considered.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 4134292
zbMATH Keywords: tree class
zbMATH Keywords: connected component
zbMATH Keywords: stable Auslander-Reiten quiver
zbMATH Keywords: group algebra
zbMATH Keywords: Dynkin diagram
zbMATH Keywords: Euclidean diagram
zbMATH Keywords: block
zbMATH Keywords: Klein four group
zbMATH Keywords: defect group
MaRDI profile type: Publication
cites work: Q3722697
cites work: Modular representation theory: New trends and methods
cites work: Notes on Representations of Finite Groups, I
cites work: Auslander-reiten sequences with few middle terms and applications to string algebrass
cites work: Blocks whose defect groups are Klein four groups: A correction
cites work: Algebras and Dihedral Defect Groups
cites work: On the vertices of modules in the Auslander-Reiten quiver of p-groups
cites work: Functors on categories of finite group representations
cites work: Q3890860
cites work: Module correspondence in Auslander-Reiten quivers for finite groups
cites work: The Auslander-Reiten quiver of a finite group
cites work: On the Auslander-Reiten quiver of a finite group
description: scientific article; zbMATH DE number 4134292