Schubert polynomial expansions revisited

TL;DR

This paper presents an elementary, nonnegative expansion method for Schubert polynomials using divided difference formalism, recovering known expansions and introducing new properties of slide polynomials.

Contribution

It introduces a simple, elementary approach to Schubert polynomial expansions, unifies existing methods, and extends divided difference formalism to slide and forest polynomials.

Findings

Recovered pipe dream and slide polynomial expansions

Established divided difference formalism for slide polynomials

Provided a method to extract coefficients in polynomial decompositions

Abstract

We give an elementary approach utilizing only the divided difference formalism for obtaining expansions of Schubert polynomials that are manifestly nonnegative, by studying solutions to the equation $\sum Y_i\partial_i=\mathrm{id}$ on polynomials with no constant term. This in particular recovers the pipe dream and slide polynomial expansions. We also show that slide polynomials satisfy an analogue of the divided difference formalisms for Schubert polynomials and forest polynomials, which gives a simple method for extracting the coefficients of slide polynomials in the slide polynomial decomposition of an arbitrary polynomial.