A derived equivalence for blocks with dihedral defect groups
type: Item
instance of: scholarly article
title: A derived equivalence for blocks with dihedral defect groups
zbMATH Open document ID: 0805.20008
author: Markus Linckelmann
published in: Journal of Algebra
publication date: 1995-02-07T00:00:00Z
review text: Let \(G\) be a finite group, \(k\) be an algebraically closed field of positive characteristic \(p\). The term ``block'' means a primitive idempotent of the center of \(kG\). If \(b\) is a block, the author calls the group \(N_ G(P_ \gamma)/PC_ G(P)\) internal quotient of \(b\), where \(P_ \gamma\) is a maximal local pointed group contained in \(G\) in Puig's terminology. Two blocks \(b_ 1\) and \(b_ 2\) of the groups \(G_ 1\) and \(G_ 2\) are derived equivalent if the algebras \(kG_ 1b_ 1\) and \(kG_ 2b_ 2\) are so. The main result of the paper is the following Theorem: Any two blocks with a common dihedral defect group having three isomorphism classes of simple modules are derived equivalent. In particular, for a defect group which is a Klein four group the author obtains the following Corollary: If \(b\) is a block of a finite group \(G\) having a Klein four group \(P\) as a defect group, then \(kGb\) and \(k(P \rtimes E)\) are derived equivalent, where \(P \rtimes E\) is the semidirect product of \(P\) by the internal quotient \(E\) of \(b\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 563137
zbMATH Keywords: derived equivalences
zbMATH Keywords: Morita equivalence
zbMATH Keywords: finite group
zbMATH Keywords: primitive idempotent
zbMATH Keywords: center
zbMATH Keywords: internal quotient
zbMATH Keywords: maximal local pointed group
zbMATH Keywords: derived equivalent
zbMATH Keywords: blocks
zbMATH Keywords: dihedral defect group
MaRDI profile type: Publication
full work available at URL: jabr.1994.1061
OpenAlex ID: W2035004644
DOI: 10.1006/JABR.1994.1061
description: scientific article; zbMATH DE number 563137