Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams

TL;DR

This paper introduces new polynomial bases and combinatorial rules that refine the understanding of Schubert polynomials, Stanley symmetric functions, and their expansions, providing positive combinatorial formulas and structure constants.

Contribution

It defines quasi-Yamanouchi pipe dreams and new bases for polynomials, offering combinatorial rules for expansions and structure constants of Schubert polynomials and Stanley symmetric functions.

Findings

New bases for polynomials lifting quasisymmetric functions.

Combinatorial rules for Schubert polynomial expansions.

Formulas for products of Stanley symmetric functions.

Abstract

We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize the quasi-shuffle product and shuffle product, respectively. We use this to give a Littlewood--Richardson rule for expanding a product of Schubert polynomials into fundamental slide polynomials and to give formulas for products of Stanley symmetric functions in terms of Schubert structure constants.