Non-commutative Pieri operators on posets

TL;DR

This paper introduces a framework using noncommutative symmetric functions to generate quasi-symmetric functions from graded posets, unifying various algebraic combinatorics approaches and linking different Hopf algebras.

Contribution

It develops a new representation-theoretic construction connecting noncommutative symmetric functions with quasi-symmetric functions via poset intervals.

Findings

Establishes a Hopf morphism from poset intervals to quasi-symmetric functions.

Shows duality between Billera-Liu algebra and Stembridge's peak algebra.

Connects these algebraic structures to Schubert calculus for isotropic flag manifolds.

Abstract

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.