Irreducible modules for classical and alternating groups.
type: Item
instance of: scholarly article
title: Irreducible modules for classical and alternating groups.
zbMATH Open document ID: 1124.20305
author: William J. Husen
published in: Journal of Algebra
publication date: 2000-05-17T00:00:00Z
review text: From the introduction: In this article we consider the following question: Given finite groups \(H<G\) and an absolutely irreducible \(\mathbb{F} G\)-module \(M\), when does \(M\!\!\downarrow_H\) remain irreducible? The question arises naturally in the study of the maximal subgroups of the classical groups. We assume that \(H\cong A_n\) or \(2.A_n\), \(n\geq 15\) and that \(G\) is a simply connected classical group over \(\mathbb{F}\), where \(\mathbb{F}\) is an algebraically closed field of characteristic \(p\geq 0\). In particular \(G=\text{SL}(V)\), \(\text{Sp}(V)\) or \(\text{Spin}(V)\) where \(V\) be a vector space over \(\mathbb{F}\). For convenience, we assume that \(\dim(V)\) is even whenever \(G=\text{Spin}(V)\) and \(p=2\). In this context we prove the following: Theorem 1.1. Suppose that \(M=M(\lambda)\) is a tensor indecomposable \(\mathbb{F} G\)-module such that \(M\!\!\downarrow_H\) is absolutely irreducible and \(\dim(M)>\dim(V)\). 1. If \(G=\text{SL}(V)\), then either (a) \(\dim(V)\leq n^3\) or (b) \(\lambda\) or \(\rho(\lambda)=3\lambda_1\), \(2\lambda_1+\lambda_2\), \(2\lambda_1+\lambda_\ell\), \(\lambda_2+\lambda_\ell\) or \(\lambda_3\) and \(\dim(V)\leq n^4\). 2. If \(G=\text{Sp}(V)\) or \(\text{Spin}(V)\), then either (a) \(\dim(V)\leq 2n^4\) or (b) \(\lambda=3\lambda_1\), \(2\lambda_1+\lambda_2\) or \(\lambda_3\) and \(\dim(V)\leq 2n^6\). Furthermore, if \(n\geq 74\) and \(p\neq 2\), then \(G\neq\text{Sp}(V)\). -- This result strengthens the main result of the author [Pac. J. Math. 192, No. 2, 297-306 (2000; Zbl 1009.20053)].
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1448621
zbMATH Keywords: finite groups
zbMATH Keywords: absolutely irreducible modules
zbMATH Keywords: maximal subgroups of classical groups
zbMATH Keywords: simply connected classical groups
zbMATH Keywords: tensor indecomposable modules
zbMATH Keywords: irreducible restrictions
MaRDI profile type: Publication
full work available at URL: jabr.1999.8213
OpenAlex ID: W2031906275
cites work: Q4002278
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Restrictions of \(\Omega_m(q)\)-modules to alternating groups
cites work: Irreducibility of alternating and symmetric squares
cites work: On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups.
cites work: WEIGHTS OF INFINITESIMALLY IRREDUCIBLE REPRESENTATIONS OF CHEVALLEY GROUPS OVER A FIELD OF PRIME CHARACTERISTIC
cites work: Q3663490
cites work: An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field
DOI: 10.1006/JABR.1999.8213
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description: scientific article; zbMATH DE number 1448621