On a Conjecture of Foulkes

Suppose that Ω = {1, 2,…,ab} for some non-negative integers a and b. Denote by P(a, b) the set of unordered partitions of Ω into a parts of cardinality b. In this paper we study the decomposition of the permutation module CP(a, b) where C is the field of complex numbers. In particular, we show that CP(3, b) is isomorphic to a submodule of CP(b, 3) for b ≥ 3. This settles the next unproven case of a conjecture of Foulkes.