Some results related to the slide decomposition of Schubert polynomials

TL;DR

This paper explores new combinatorial and topological methods to analyze Schubert polynomial expansions, establishing connections with subword complexes, and providing tableau-based definitions and geometric insights.

Contribution

It extends subword complex methods to more general expansions of Schubert polynomials and introduces tableau-based definitions for related polynomials.

Findings

Subword complex is a ball or a sphere.

Expansion into forest polynomials corresponds to sub-balls in the subword complex.

Slide polynomials are not Gr"obner degenerations of matrix Schubert varieties.

Abstract

The expansion of a Schubert polynomial into slide polynomials corresponds to a sum over sub-balls in the subword complex. There has been recent interest in other, coarser, expansions of Schubert polynomials. We extend the methods used in [KM04] to prove that the subword complex is a ball or a sphere to a more general method, and use it to prove that the expansion of a Schubert polynomial into forest polynomials also corresponds to a sum over sub-balls in the subword complex. When expanding the product $\mathfrak{S}_\pi\mathfrak{S}_\rho$ of two Schubert polynomials into Schubert polynomials $\mathfrak{S}_\sigma$, there is a bijection between shuffles of reduced words for $\pi$ and $\rho$ and reduced words for $\sigma$ (counted with multiplicity). We give such a bijection for Monk's rule and Sottile's Pieri rule. We give tableau-based definitions of slide polynomials, glide polynomials, and fundamental quasisymmetric polynomials and show that the expansion of a Schur polynomial into fundamental quasisymmetric polynomials corresponds to a sum over sub-balls in the tableau complex. The Schubert polynomials are the cohomology classes of matrix Schubert varieties, but there is no geometric explanation of the slide polynomials. We show that the slide polynomials are not (antidiagonal) Gr\"obner degenerations of matrix Schubert varieties, answering in the negative a question of [ST21, Section 1.4].