Generalized Foulkes' Conjecture and tableaux construction

Foulkes conjectured that for n = ab and a less than or equal to b, every irreducible module occurring as a constituent in 1(Sb,Sa)(Sn) occurs with greater or equal multiplicity in 1(Sa,Sb)(Sn). We generalize this to say those irreducibles also occur in 1(Sd,Sc)(Sn), where cd = n and c, d greater than or equal to 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 S-b, S-a modules and in doing so, completely classify all such tableaux for a = 2 and a = 3. (C) 2004 Elsevier Inc. All rights reserved.