Divided symmetrization and quasisymmetric functions

TL;DR

This paper explores the relationship between quasisymmetric polynomials and the divided symmetrization operator, providing new formulas and combinatorial insights relevant to Schubert calculus and volume polynomials of permutahedra.

Contribution

It introduces a method to compute divided symmetrization of quasisymmetric polynomials and connects it to a direct sum decomposition involving quasisymmetric ideals.

Findings

Divided symmetrization of quasisymmetric polynomials can be explicitly computed.

Provides combinatorial examples illustrating the computation.

Establishes a natural computation framework using a decomposition related to quasisymmetric ideals.

Abstract

Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided symmetrization is a linear form which acts on the space of polynomials in $n$ indeterminates of degree $n-1$. We first show that divided symmetrization applied to a quasisymmetric polynomial in $m$ indeterminates can be easily determined. Several examples with a strong combinatorial flavor are given. Then, we prove that the divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to Aval-Bergeron-Bergeron involving the ideal generated by positive degree quasisymmetric polynomials in $n$ indeterminates.