Hopf Algebras and Edge-labeled Posets

Given a nite graded poset with labeled Hasse diagram, we construct a quasi-symmetric generating function for chains whose labels have xed descents. This is a common generalization of a generating function for the ag f-vector de-ned by Ehrenborg and of a symmetric function associated to certain edge-labeled posets which arose in the theory of Schubert polynomials. We show this construction gives a Hopf morphism from an incidence Hopf algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions. Joni and Rota 8], and later Schmitt 13] construct Hopf algebras from partially ordered sets, giving a global algebraic framework for studying partially ordered sets. For every graded partially ordered set, Ehrenborg 4] deenes a quasi-symmetric generating function for its ag f-vector. He shows that this induces a Hopf morphism from the Hopf algebra of graded posets to the Hopf algebra of quasi-symmetric functions. Edge-labeled posets are nite graded partially ordered sets, the edges of the Hasse diagrams of which are labeled with integers. Following the construction of Stanley's symmetric function 14], we associate with each such poset a quasi-symmetric generating function for maximal chains whose sequence of edge labels has xed descents. We show that this reduces to Ehrenborg's function in an important special case and induces a Hopf morphism from the Hopf algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions. While studying structure constants for Schubert polynomials, we deened a symmetric function for any edge-labeled poset with a certain symmetry 3], giving a uniied construction of skew Schur functions, Stanley symmetric functions, and skew Schu-bert functions. We show that this symmetric function equals the quasi-symmetric generating function deened here.