QSym over Sym has a stable basis

TL;DR

This paper proves a conjecture that a specific subset of quasisymmetric polynomials forms a basis for the coinvariant space, providing a constructive proof of a fundamental module structure result.

Contribution

It establishes that the conjectured basis of quasisymmetric polynomials is valid, confirming the free module structure over symmetric polynomials with a constructive approach.

Findings

Confirmed the basis conjecture for quasisymmetric polynomials

Provided a constructive proof of the Garsia-Wallach result

Showed that quasisymmetric polynomials form a free module of dimension n!

Abstract

We prove that the subset of quasisymmetric polynomials conjectured by Bergeron and Reutenauer to be a basis for the coinvariant space of quasisymmetric polynomials is indeed a basis. This provides the first constructive proof of the Garsia-Wallach result stating that quasisymmetric polynomials form a free module over symmetric polynomials and that the dimension of this module is n!.