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Irreducibility of alternating and symmetric squares. (English) Zbl 0919.20009
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 G. D. James, Math. Proc. Camb. Philos. Soc. 94, 417-424 (1983; Zbl 0544.20011), and A. Wagner, Arch. Math. 29, 583-589 (1977; Zbl 0383.20009)] allows one to rule out the remaining cases.

20C33 Representations of finite groups of Lie type
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C20 Modular representations and characters
20E28 Maximal subgroups
20C30 Representations of finite symmetric groups
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