Derivatives of padded Schubert polynomials through pipe dreams

TL;DR

This paper introduces a combinatorial approach using pipe dreams to prove positivity results for derivatives of padded Schubert polynomials, extending previous work on differential operators in algebraic combinatorics.

Contribution

It provides a new combinatorial proof for the positivity of derivatives of extit{ extbf{padded}} Schubert polynomials for dominant permutations, expanding the understanding of their algebraic structure.

Findings

Established a combinatorial proof using pipe dreams

Extended positivity results to extit{ extbf{padded}} Schubert polynomials

Connected differential operators with combinatorial models

Abstract

In recent work, Hamaker, Pechenik, Speyer, and Weigandt showed that a certain differential operator $\nabla$ expands positively in the basis of Schubert polynomials. For $\pi$ a dominant permutation, Gaetz and Tung showed an analogous positivity result holds for a dual differential operator $\Delta$ acting on $\pi$-padded Schubert polynomials. In this paper, we provide a new proof of the result of Gaetz and Tung using the combinatorics of pipe dreams.