On conjectures of Foulkes, Siemons and Wagner and Stanley

Abstract

Let A = (AI, ... ,Ar ) be a partition of n. An unordered A-tabloid is a partition of the set {I, 2, ... , n} into r pairwise disjoint sets of sizes AI, ,Ar . Let F denote the field of complex numbers and C the symmetric group of {I, 2, , n}. Define HA to be the permutation module of FC whose basis is the set of unordered A-tabloids. Foulkes conjectured in [13] that there exists an injective FC-homomorphism H(b a ) -t H(a b ) when a ::; b. Independently Siemons and Wagner [27] and Stanley [29] generalized this conjecture to ask if there exists an injective map HA -t HA'. In this thesis we investigate these conjectures.EThOS - Electronic Theses Online ServiceGBUnited Kingdo