Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity

TL;DR

This paper introduces dual equivalence graphs, axiomatizes them, and uses them to provide a combinatorial proof of the symmetry and Schur positivity of certain symmetric functions, including Macdonald polynomials.

Contribution

It develops a new combinatorial framework called dual equivalence graphs and applies it to prove key properties of Macdonald and ribbon tableau generating functions.

Findings

Proves symmetry and Schur positivity of ribbon tableau generating functions

Provides a combinatorial formula for Macdonald polynomial Schur expansion

Establishes dual equivalence graphs as a tool for symmetric function 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. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomials.