Quasisymmetric divided differences

TL;DR

This paper introduces a quasisymmetric analogue of Schubert polynomial theory using forest polynomials and new operators, extending to colored functions and resolving a key conjecture in quasisymmetric harmonics.

Contribution

It develops a novel framework with forest polynomials and operators for quasisymmetric functions, extending classical theories and solving a significant conjecture.

Findings

Introduction of forest polynomials as quasisymmetric counterparts to Schubert polynomials

Extension of the theory to m-colored quasisymmetric functions

Resolution of a conjecture on quasisymmetric harmonics

Abstract

We develop a quasisymmetric analogue of the combinatorial theory of Schubert polynomials and the associated divided difference operators. Our counterparts are "forest polynomials", and a new family of linear operators, whose theory of compositions is governed by forests and the "Thompson monoid". Our approach extends naturally to $m$-colored quasisymmetric functions. We then give several applications of our theory to fundamental quasisymmetric functions, the study of quasisymmetric coinvariant rings and their associated harmonics, and positivity results for various expansions. In particular we resolve a conjecture of Aval-Bergeron-Li regarding quasisymmetric harmonics.