Noncommutative Pieri Operators on Posets

We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P . The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge related to enriched P-partitions, and connect this to the combinatorics of the Schubert calculus for isotropic flag manifolds. Dedicated to the memory of Dr. Gian-Carlo Rota, who inspired us to seek the algebraic structures underlying combinatorics.