# A matroid-friendly basis for the quasisymmetric functions

## TL;DR

This paper introduces a new basis for quasisymmetric functions with positive structure constants, revealing deep connections to matroid theory and answering a key open question about the Hilbert basis.

## Contribution

It presents a novel Z-basis for QSym with nonnegative structure constants and explores its properties related to matroids, including injectivity and decomposability insights.

## Key findings

- New basis for QSym with nonnegative structure constants
- Injectivity of the morphism on rank two matroids
- Decomposability of quasisymmetric functions mirrors matroid base polytope decomposability

## Abstract

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer is given to the Hilbert basis question raised by Billera, Jia, and Reiner.