Derivatives of padded Schubert polynomials through pipe dreams

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.