On primitive linear representations of finite groups
type: Item
instance of: scholarly article
title: On primitive linear representations of finite groups
zbMATH Open document ID: 0991.20008
author: Michael Aschbacher
published in: Journal of Algebra
publication date: 2002-05-15T00:00:00Z
review text: A representation \(\pi\) of a finite group \(G\) over a field \(F\) is called primitive if it is irreducible and it is not induced from any representation of a proper subgroup of \(G\). An application of Clifford's theorem shows that a primitive representation is always quasi-primitive, that is, its restriction to each normal subgroup of \(G\) is homogeneous. Given two representations \(\pi_1\) and \(\pi_2\) of groups \(G_1\) and \(G_2\), their tensor product \(\pi_1\otimes\pi_2\) is defined, and is a representation of the direct product \(G_1\times G_2\). The two main results of the paper, in the special case where \(F\) is the field of complex numbers, show that if \(\pi_1\) and \(\pi_2\) are primitive (resp.~irreducible and quasi-primitive), then \(\pi_1\otimes\pi_2\) is primitive (resp.~irreducible and quasi-primitive). However, the results are stated and proved over arbitrary fields, where some extra assumption on the representations is needed. A representation \(\pi\) of a group \(G\) over \(F\) is said to be AI if \(\pi\) is completely reducible and all irreducible subrepresentations of all restrictions of \(\pi\) to normal subgroups of \(G\) are absolutely irreducible. The two main results cited above remain true over arbitrary fields provided one adds the hypothesis that \(\pi_1\) and \(\pi_2\) are AI; one then has the additional conclusion that \(\pi_1\otimes\pi_2\) is AI. The proof of one of the results (the one about primitive representations) requires a weak appeal to the classification of finite simple groups, but according to the author it may be possible to avoid that appeal with the right observation or some extra work.
reviewed by: Sandro Mattarei
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1565878
zbMATH Keywords: quasi-primitive representations
zbMATH Keywords: primitive representations
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible representations
zbMATH Keywords: tensor products
MaRDI profile type: Publication
full work available at URL: jabr.2000.8532
OpenAlex ID: W2054086017
cites work: Maximal subgroups of finite groups
cites work: Primitive solvable groups
cites work: Q5340151
cites work: Q3708174
DOI: 10.1006/JABR.2000.8532
description: scientific article; zbMATH DE number 1565878