Slide polynomials and subword complexes

TL;DR

This paper introduces slide complexes, a new decomposition of subword complexes related to Schubert polynomials, showing they are topologically simple, which advances understanding of their combinatorial and geometric structure.

Contribution

It defines slide complexes as a new stratification of subword complexes, linking them to slide polynomials and topologically characterizing them as balls or spheres.

Findings

Slide complexes correspond to slide polynomials.

They are homeomorphic to balls or spheres.

This decomposition enhances understanding of subword complexes' structure.

Abstract

Subword complexes were defined by A.Knutson and E.Miller in 2004 for describing Gr\"obner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S.Assaf and D.Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres.