P-partition products and fundamental quasi-symmetric function positivity

Abstract

We show that certain differences of products K Q ∧ R , θ K Q ∨ R , θ − K Q , θ K R , θ of P-partition generating functions are positive in the basis of fundamental quasi-symmetric functions L α . This result interpolates between recent Schur positivity and monomial positivity results of the same flavor. We study the case of chains in detail, introducing certain “cell transfer” operations on compositions and a related “ L-positivity” poset. We introduce and study quasi-symmetric functions called wave Schur functions and use them to establish, in the case of chains, that K Q ∧ R , θ K Q ∨ R , θ − K Q , θ K R , θ is itself equal to a single generating function K P , θ for a labeled poset ( P , θ ) . In the course of our investigations we establish some factorization properties of the ring QSym of quasi-symmetric functions.