Dual equivalence and Schur positivity

TL;DR

This paper introduces dual equivalence for combinatorial objects with descent sets, providing a framework that proves symmetry and Schur positivity of their generating functions, with explicit Schur expansion formulas.

Contribution

It formalizes dual equivalence for combinatorial objects and simplifies proofs of symmetry and Schur positivity, including explicit Schur expansion formulas.

Findings

Dual equivalence establishes symmetry and Schur positivity.

Explicit Schur expansion formulas are derived.

Simplified proofs in cases with nice reading words.

Abstract

We define dual equivalence for any collection of combinatorial objects endowed with a descent set, and we show that giving a dual equivalence establishes the symmetry and Schur positivity of the quasi-symmetric generating function. We give an explicit formula for the Schur expansion of the generating function in terms of distinguished elements of the dual equivalence classes. These concepts and proofs simplify in the ubiquitous case when the collection of objects has a sufficiently nice reading word.