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P-partition products and fundamental quasi-symmetric function positivity

Thomas Lam·Pavlo Pylyavskyy
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Abstract

We show that certain differences of products 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 an interesting 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 the difference of products we study 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 of quasisymmetric functions.

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Thomas Lam, Pavlo Pylyavskyy. 2006-09-08. P-partition products and fundamental quasi-symmetric function positivity. https://arxiv.org/abs/math/0609249

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