On conjugacy classes of \(S_{n}\) containing all irreducibles (Q1650038)

For any integer \(n \geq 2\) and any partition \(\lambda\) of \(n\), the permutations in the symmetric group \(S_n\) whose cycle type is \(\lambda\) form a conjugacy class \(K_\lambda\) of \(S_n\). The group \(S_n\) acts on this \(K_\lambda\) by conjugation; this action induces a permutation module \(P_\lambda\). The author proves that for each \(n \geq 8\) and also for \(n = 6\) (but not for any other \(n\)), there exists some partition \(\lambda\) of \(n\) such that the \(S_n\)-module \(P_\lambda\) contains every irreducible \(S_n\)-module. In group-theoretical language, this means that \(K_\lambda\) is a ``global class'' for \(S_n\). The author's proof is based on plethysms of symmetric functions, Schur-positivity results, intricate Young-diagrammatic constructions (proving that certain Littlewood-Richardson coefficients are positive), and a number-theoretical result by \textit{R. E. Dressler} [Proc. Am. Math. Soc. 33, 226--228 (1972; Zbl 0236.10030)] saying that every positive integer \(> 9\) is a sum of distinct odd primes. For \(n \geq 10\), she picks her \(\lambda\) to be a partition of \(n\) into distinct odd parts such that each part is prime or \(1\), and such that at least two parts are \(> 1\).