A counter-example to the admissibility of the \(\gamma\)-filtration on 2- groups
type: Item
instance of: scholarly article
title: A counter-example to the admissibility of the \(\gamma\)-filtration on 2- groups
zbMATH Open document ID: 0836.55011
author: Sally Cockburn
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: The main result in this paper is that Atiyah's conjecture on the equivalence of the topological and \(\gamma\)-filtrations on the representation ring \(R(G)\) fails when \(G = D^r\), \(r\geq 3\), the central product of \(r\) copies of the dihedral group of order 8 (extra special). The method is to show that the \(\gamma\)-filtration is not `admissible' in the sense that it behaves badly with respect to induction. In passing it is also shown that for \(r = 1\) or 2 we do have admissibility. As with all the counterexamples now known, the filtrations are equivalent infinitely often, i.e. \(R^\gamma_{2k} = R^{\text{top}}_{2k}\), provided certain congruence conditions on \(k\) are satisfied. It would be good to have a systematic explanation of this phenomenon.
Mathematics Subject Classification ID: 55T05
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 614221
zbMATH Keywords: gamma filtration
zbMATH Keywords: topological filtration
zbMATH Keywords: central product of \(r\) copies of the dihedral group of order 8
zbMATH Keywords: \(\gamma\)-filtration
zbMATH Keywords: induction
reviewed by: Charles B. Thomas
MaRDI profile type: Publication
full work available at URL: jabr.1994.1198
OpenAlex ID: W1988545382
DOI: 10.1006/JABR.1994.1198
description: scientific article; zbMATH DE number 614221