Absolute irreducibility for finitary linear groups
type: Item
instance of: scholarly article
title: Absolute irreducibility for finitary linear groups
zbMATH Open document ID: 0827.20057
author: Felix Leinen
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1995-12-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108343
review text: In this short note the author contributes to the present rapid development of the theory of finitary linear groups by considering absolute irreducibility. Let \(V\) be a vector space over the field \(K\) and \(L\) an extension field of \(K\). Then \(\text{GL}(V)\) acts in the obvious way on \(\text{GL}(V^L)\) for \(V^L=L\otimes_KV\). Call a subgroup \(G\) of \(\text{GL}(V)\) absolutely irreducible if \(V^L\) is irreducible as \(LG\)-module for every extension field \(L\) of \(K\). The author proves the following, thus extending the finite-dimensional (linear) case. Theorem. Let \(V\) be a vector space over the field \(K\) and let \(G\) be an irreducible subgroup of \(\text{FGL}(V)\). Then there exists a finite field extension \(L/K\) such that \(V\) is also a vector space over \(L\), and such that \(V\) is absolutely irreducible as \(LG\)-module. Here the degree of the field extension \(L/K\) is finite and divides the \(K\)-dimension of every finite-dimensional \(L\)-subspace of \(V\). Examples of \(L\)-subspaces are \(C_V(F)\) and \([V,F]\) for every finitely generated subgroup \(F\) of \(G\). Slightly surprising perhaps is the finiteness of the degree \(L/K\). In the finite-dimensional case, of course, this is obviously finite.
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 763292
zbMATH Keywords: absolutely irreducible modules
zbMATH Keywords: finitary linear groups
zbMATH Keywords: absolute irreducibility
zbMATH Keywords: irreducible subgroups
zbMATH Keywords: finite field extensions
zbMATH Keywords: finitely generated subgroups
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
cites work: Q5569204
cites work: Q5676405
description: scientific article; zbMATH DE number 763292