On Auslander-Reiten components and height zero lattices for integral group rings.
type: Item
instance of: scholarly article
title: On Auslander-Reiten components and height zero lattices for integral group rings.
zbMATH Open document ID: 1318.20006
author: Shigeto Kawata
published in: Algebras and Representation Theory
publication date: 2014-11-17T00:00:00Z
review text: Let \(G\) be a finite group, let \(k=k'\) be an algebraically closed field of characteristic \(p>0\), and let \((K,R,k)\) and \((K',R',k')\) be \(p\)-local systems such that \(R\) is a ramified extension of \(R'\) of ramification index at least \(3\). Then the author studies the possible structures of Auslander-Reiten components of \(RG\)-lattices \(L\) of height \(0\). Webb showed that only components of tree class \(A_\infty\), \(B_\infty\), \(C_\infty\), \(D_\infty\) and \(A_\infty^\infty\) or a Euclidean diagram can occur. The paper under review shows that in case of tree class \(A_\infty\) an indecomposable \(RG\)-lattice of height \(0\) lies at basically the same position as its source. Moreover, the author gives various criteria when the tree class of a component is \(A_\infty\). One of the main tools for the proof is Knörr's concept of virtually irreducible lattices.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 6370054
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: integral representations of finite groups
zbMATH Keywords: height zero lattices
zbMATH Keywords: integral group rings
zbMATH Keywords: virtually irreducible lattices
MaRDI profile type: Publication
full work available at URL: s10468-013-9462-7
OpenAlex ID: W4246227809
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DOI: 10.1007/S10468-013-9462-7
description: scientific article; zbMATH DE number 6370054