Nonnegative Hall Polynomials

Abstract

The number of subgroups of type μ and cotype ν in a finite abelian p-group of type λ is a polynomialg\(_{\mu v}^\lambda (p)\)with integral coefficients. We prove g\(_{\mu v}^\lambda (p)\) has nonnegative coefficients for all partitions μ and ν if and only if no two parts of λ differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections ϕ that associate to each subgroup a vector dominated componentwise by λ. The nonzero components of ϕ(H) are the parts of μ, the type of H; if no two parts of λ differ by more than one, the nonzero components of λ − ϕ(H) are the parts of ν, the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.