Relative higher relation modules, and a property of irreducible K- representations of \(GL_ n(K)\)
type: Item
instance of: scholarly article
title: Relative higher relation modules, and a property of irreducible K- representations of \(GL_ n(K)\)
zbMATH Open document ID: 0632.20005
DOI: 10.1016/0021-8693(86)90241-3
author: Brian Hartley
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: The object of the paper is the persistence of free submodules for finite subgroups of \(GL_ n(K)\) in the following sense. Let K be an arbitrary field, and V be a finite-dimensional irreducible (left) module for \(GL_ n(K)\) over K. If K is infinite, assume that V affords a polynomial representation of \(GL_ n(K)\). Let E be the natural left \(GL_ n(K)\)- module and G be a nontrivial finite subgroup of \(GL_ n(K)\). Suppose that \(E\cong KG\oplus E_ 1\), as KG-modules, for some KG-module \(E_ 1\). Then the main result says that one of the following holds. (a) V contains a nonzero free KG-submodule; (b) \(\dim_ K(V)=1\); (c) \(E_ 1=0\), G is an elementary abelian 2-group of order \(>2\), the structure of V is explicitly given and V contains no nontrivial free KG-submodule. Using standad facts the proof of the main result is reduced to the case when K is infinite. The resulting special case corrects theorem 3.3 of a joint paper of the author and \textit{A. I. Lichtman} [J. Pure Appl. Algebra 22, 75-89 (1981; Zbl 0467.20009)]. Finally using results of Borel and Tits a similar theorem to the main result is proved when K is infinite but V does not necessarily afford a polynomial representation of \(GL_ n(K)\). The connection to the study of relative higher relation modules is given [cf. loc. cit.].
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 4027732
zbMATH Keywords: free submodules
zbMATH Keywords: finite subgroups of \(GL_ n(K)\)
zbMATH Keywords: polynomial representation
zbMATH Keywords: relative higher relation modules
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90241-3
OpenAlex ID: W2060677367
cites work: Homomorphismes ''abstraits'' de groupes algébriques simples
cites work: Polynomial representations of \(GL_n\)
cites work: Relative higher relation modules
cites work: Q3934684
cites work: Q3952291
description: scientific article; zbMATH DE number 4027732