On certain quotients of the Green rings of dihedral 2-groups.
type: Item
instance of: scholarly article
title: On certain quotients of the Green rings of dihedral 2-groups.
zbMATH Open document ID: 1152.20002
author: Louise Archer
published in: Journal of Pure and Applied Algebra
publication date: 2008-06-05T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic 2, let \(G\) be a dihedral 2-group such that \(|G|\geq 8\), and let \(A(kG)\) be the Green ring of the group algebra \(kG\). By a theorem of Benson and Carlson, the isomorphism classes of the even-dimensional indecomposable \(kG\)-modules span an ideal \(A(kG;2)\) of \(A(kG)\), and 0 is the only nilpotent element in \(\overline A(kG):=A(kG)/A(kG;2)\). The author shows that the images of the isomorphism classes of the odd-dimensional indecomposable \(kG\)-modules in \(\overline A(kG)\) form an Abelian group \(\Gamma\) which is torsion-free and infinitely generated. The precise nature of \(\Gamma\) remains open. The author shows, however, that the multiplication in \(\Gamma\) induces graph automorphisms between components of the Auslander-Reiten quiver of \(kG\).
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 5285326
zbMATH Keywords: Green rings
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: almost split sequences
zbMATH Keywords: band modules
zbMATH Keywords: string modules
zbMATH Keywords: endo-trivial modules
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: j.jpaa.2007.11.012
OpenAlex ID: W2061207470
cites work: Almost-split sequences and group rings
cites work: Modular representation theory: New trends and methods
cites work: Q3027209
cites work: Q3758103
cites work: Nilpotent elements in the Green ring
cites work: Q4012104
cites work: Torsion endo-trivial modules
cites work: Certain representation algebras
cites work: The modular representation algebra of a finite group
cites work: Nilpotent elements of order \(3\) in the Green ring of the dihedral-\(2\)- groups
cites work: The indecomposable representations of the dihedral 2-groups
cites work: Q4749976
cites work: Nilpotent elements in representation rings
cites work: Nilpotent elements in representation rings over fields of characteristic 2
DOI: 10.1016/J.JPAA.2007.11.012
description: scientific article; zbMATH DE number 5285326