On a conjecture of Foulkes

Suppose that Omega = {1, 2,..., ab} for some non-negative integers a and b. Denote hv P(a, b) the set of unordered partitions of Omega into a parts of cardinality b. In this paper we study the decomposition of the permutation module CP(3, b) where C is the field of complex numbers. in particular, we show that CP(b,3) bf is isomorphic to a submodule of CP(b,3) for b greater than or equal to 3. This settles the next unproven case of a conjecture of Foulkes. (C) 2000 Academic Press.