Some Computations Regarding Foulkes' Conjecture

We describe how certain permutation actions of large symmetric groups can be efficiently implemented on a computer. Using a specially tailored adaptation of a general technique to enumerate huge orbits, and substantial distributed computation on a cluster of workstations, we collect further evidence related to the approach to Foulkes’ conjecture suggested in [Black and List, 1989]. 1 Foulkes’ conjecture To state Foulkes’ conjecture we first introduce some notation. Let N be the set of positive integers, let Q be the set of rational numbers, and denote by Mn := {1, 2, 3, . . . , n} for n ∈ N the set of natural numbers less than or equal to n. We denote the symmetric group on n points by Sn := {π : Mn → Mn | π bijective}, with concatenation of maps as product, which we denote as π ◦ φ meaning “apply first φ, then π”. Form,n ∈ N let SmoSn be the wreath product of Sm and Sn, which is a semidirect product of the n-fold direct product Sn m := Sm×· · ·×Sm of copies of Sm and Sn, where the latter acts on the first by permuting the direct factors. Note that Sn m can be identified with the set of maps {f : Mn → Sm}. Hence, Sm o Sn = Sn m o Sn with product (f, π) · (f ′, π′) := (f · (f ′ ◦ π−1), π ◦ π′), where we multiply maps f : Mn → Sm pointwise using the product in Sm. The wreath product Sm o Sn has order |Sm o Sn| = (m!)n · n!, and embeds into Smn by letting the i-th direct factor of Sn m, for i = 1, . . . , n, permute the points {(i− 1)m+ 1, . . . , im} and keep all other points in Mmn fixed, while Sn acts on Mmn by permuting these n blocks; for more details see [James and Kerber, 1981, Section 4.1]. We denote by Ωm,n the set {(Sm oSn) ◦ π | π ∈ Smn} of right cosets of Sm oSn in Smn, and by QΩm,n the associated permutation right QSmn-module. It is easily seen by an induction argument that for m ≥ n we have |Sn o Sm| ≤ |Sm o Sn|. Thus we have |Ωm,n| ≤ |Ωn,m|. But in fact much more is conjectured to be true: 1.1 Conjecture ([Foulkes, 1950]) Let m,n ∈ N with m ≥ n. Then the permutation module QΩm,n is a QSmn-submodule of the permutation module QΩn,m. An outline of this note is as follows: In Section 2 we describe how the action of Smn on Ωm,n can be efficiently implemented on a computer. This implementation will be used for calculations connected to the approach to Foulkes’ conjecture suggested in [Black and List, 1989]. Our description uses the notion of Schur bases, which are introduced in Section 3, while in Section 4 the approach of Black and List is discussed. In Section 5 our particular computational techniques are explained, and in the final Section 6 actual computational results are presented. There we also describe, for which values of m and n the conjecture has been verified computationally so far.