On the irreducibility of alternating powers and symmetric squares (Q1337771)

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.