Irreducibility of alternating and symmetric squares
type: Item
instance of: scholarly article
title: Irreducibility of alternating and symmetric squares
zbMATH Open document ID: 0919.20009
author: Kay Magaard
author: Gunter Malle
published in: Manuscripta Mathematica
publication date: 1999-07-06T00:00:00Z
full work available at URL: https://eudml.org/doc/156368
review text: An important problem in group theory, which has many applications, is to classify maximal subgroups of finite classical groups. In order to determine maximality of certain quasi-simple subgroups \(G\) of finite classical groups, one needs to know in particular when the (natural) largest subquotient of \(X(V)\) could be irreducible. Here \(V\) is any irreducible \(\mathbb{F} G\)-module, and \(X(V)\) is the symmetric square \(\Sigma^2(V)\) or the alternating square \(\Lambda^2(V)\) (if \(V\) is self-dual), or the tensor square \(V\otimes V^*\) (if \(V\) is not self-dual). One expects that this cannot happen in general; however it could happen for certain finite groups of Lie type over small fields as well as for certain covers of small alternating or sporadic groups. Previous results in this direction were obtained by the first author for \(G=\text{SL}_n(q)\) [Arch. Math. 63, No. 3, 211-215 (1994; Zbl 0824.20012)], and by the second author for arbitrary quasi-simple groups in the case \(\text{char}(\mathbb{F})=0\) and \(X(V)=V\otimes V^*\) [Commun. Algebra 27, No. 3, 1033-1051 (1999)]. In this paper the authors completely solve the problem in the cases where either \(G\) is a cover of the alternating group \(A_n\), or \(G=\text{SL}_n(q)\) and \(\text{char}(\mathbb{F})\) does not divide \(q\). Assume \(V\) is such that \(X(V)\) is irreducible. In the former case, it is shown that \(V\) is a subquotient of the induced representation \(1^{A_n}_{A_{n-4}}\). Similarly, in the latter case the authors show that \(V\) is a subquotient of the induced representation \(\lambda^G_P\), where \(P\) is a large parabolic subgroup of \(G\) and \(\lambda\) is a \(P\)-representation of degree \(1\). This yields an upper bound on the dimension of \(V\). Knowledge of representations of \(G\) of small dimension [see for instance \textit{G. D. James}, Math. Proc. Camb. Philos. Soc. 94, 417-424 (1983; Zbl 0544.20011), and \textit{A. Wagner}, Arch. Math. 29, 583-589 (1977; Zbl 0383.20009)] al…
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20E28
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1151661
zbMATH Keywords: special linear groups
zbMATH Keywords: alternating groups
zbMATH Keywords: alternating squares
zbMATH Keywords: symmetric squares
zbMATH Keywords: maximal subgroups
zbMATH Keywords: irreducible modules
zbMATH Keywords: finite groups of Lie type
reviewed by: Pham Huu Tiep
MaRDI profile type: Publication
OpenAlex ID: W1994138384
DOI: 10.1007/S002290050021
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description: scientific article; zbMATH DE number 1151661