Generalized Foulkesʹ Conjecture and tableaux construction

Foulkes conjectured that for n=ab and a b, every irreducible module occurring as a constituent in occurs with greater or equal multiplicity in . We generalize this to say those irreducibles also occur in , where cd=n and c,d a. We prove the generalized conjecture for a=2 and a=3 by explicitly constructing the corresponding tableaux. To do so we develop a theory of construction conditions for tableaux giving rise to modules and in doing so, completely classify all such tableaux for a=2 and a=3.