Dual equivalence graphs I: A new paradigm for Schur positivity

TL;DR

This paper introduces dual equivalence graphs, a new combinatorial framework that systematically proves symmetry and Schur positivity of quasisymmetric functions, offering a universal method for such proofs.

Contribution

It axiomatizes dual equivalence graphs and demonstrates their utility in establishing Schur positivity universally.

Findings

Dual equivalence graphs are symmetric and Schur positive.

The framework provides a universal method for proving Schur positivity.

Axiomatization of dual equivalence graphs enables systematic analysis.

Abstract

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. This provides a universal method for establishing the symmetry and Schur positivity of quasisymmetric functions.