Polynomial bases: positivity and Schur multiplication

TL;DR

This paper introduces a poset structure on various polynomial bases, proves positivity of their products, and provides a new Littlewood-Richardson rule extending previous results, with bijections connecting combinatorial models.

Contribution

It establishes a new poset framework for polynomial bases, introduces a new basis, and derives a novel Littlewood-Richardson rule for Schur and quasi-key polynomials.

Findings

Product of Schur and basis element expands positively

First Littlewood-Richardson rule for Schur and quasi-key polynomials

Bijections connect semi-skyline fillings and quasi-key tableaux

Abstract

We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions, and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters and Demazure atoms; the quasi-key, fundamental and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason and van Willigenburg's (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux.