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On the irreducibility of alternating powers and symmetric squares. (English) Zbl 0824.20012
Let \(H = \text{SL} (n,q)\), \(q > 2\), let \(F\) be a field of characteristic prime to \(q\) and \(V\) an absolutely irreducible \(FH\)-module. Let \(\text{Sym}^ k (V)\) and \(\Lambda^ k (V)\) denote the \(k\)-th symmetric and alternating (= external) power of \(V\). The author proves the following results: (1) if \(n > 4\) then \(\Lambda^ 2 (V)\) and \(\text{Sym}^ 2 (V)\) are not absolutely irreducible \(FH\)-modules; (2) If \(n > 3\) and \(3 < k < 1 + \min (q - 1, (\dim(V) /2))\) then \(\Lambda^ k(V)\) is not an absolutely irreducible \(FH\)-module.

20C33 Representations of finite groups of Lie type
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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[1] V. Landazuri andG. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. J. Algebra32, 418-443 (1974). · Zbl 0325.20008
[2] G. Seitz andA. Zalesskii, On the Minimal Degrees of Projective Representations of Finite Chevalley Groups, II. J. Algebra158, 233-243 (1993). · Zbl 0789.20014
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