Generalized Foulkes' conjecture and tableaux construction (Q1882881)

Let \(a\leq b\) be positive integers and let \(n=ab\). Let \(S_b\wr S_a\) denote the wreath product of \(S_b\) and \(S_a\), i.e. the normalizer of the Young subgroup consisting of the direct product of \(b\) copies of \(S_a\) inside \(S_{ab}\). Foulkes' conjecture states that every irreducible module occuring as a constituent in the character obtained from inducing the trivial character of \(S_b\wr S_a\) up to \(S_n\) occurs with at least the same multiplicity in the induction of the trivial character from \(S_a\wr S_b\) to \(S_n\). See \textit{H. O. Foulkes} [J. Lond. Math. Soc. 25, 205--209 (1950; Zbl 0037.14902)]. The author generalizes this conjecture to state that those irreducible modules also appear in the induction of the trivial character of \(S_d\wr S_c\) to \(S_n\) whenever \(c,d\geq a\) and \(cd=n\), and proves this generalized conjecture for \(a=2\) and \(a=3\) via tableaux. The proof is combinatorial. The author also considers versions of the Foulkes conjecture for the alternating (sign) character.