Slide polynomials and subword complexes

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.