Generalized flags in finite Abelian p-groups

Abstract

For each partition λ of n , let L λ = L λ ( p ) be the lattice of subgroups of a finite Abelian p -group of type λ. We study a topological condition necessary for the existence of an order-preserving injection of L μ into L λ . Our main result shows this topological condition is equivalent to the simple requirement that μ dominates λ. We employ results obtained by Lascoux and Schützenberger in the theory of Hall—Littlewood symmetric functions.