# Weak dual equivalence for polynomials

## TL;DR

This paper introduces weak dual equivalence, a combinatorial tool that provides new proofs of positivity properties for symmetric and Schubert polynomials, and establishes nonnegative expansion rules for key polynomials.

## Contribution

It introduces weak dual equivalence and applies it to prove positivity and expansion rules for key and Schubert polynomials, advancing combinatorial understanding.

## Key findings

- Stanley symmetric functions are Schur positive.
- Schubert polynomials are key positive.
- Nonnegative Littlewood-Richardson rule for key polynomial and Schur polynomial products.

## Abstract

We use dual equivalence to give a short, combinatorial proof that Stanley symmetric functions are Schur positive. We introduce weak dual equivalence, and use it to give a short, combinatorial proof that Schubert polynomials are key positive. To demonstrate further the utility of this new tool, we use weak dual equivalence to prove a nonnegative Littlewood--Richardson rule for the key expansion of the product of a key polynomial and a Schur polynomial, and to introduce skew key polynomials that, when skewed by a partition, expand nonnegatively in the key basis.