Abstract:
Subword complexes were defined by Knutson and Miller in 2004 to describe Gröbner degenerations of matrix Schubert varieties. Subword complexes of a certain type are called pipe dream complexes. The facets of such a complex are indexed by pipe dreams, or, equivalently, by monomials in the corresponding Schubert polynomial. In 2017 Assaf and 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. The slide complexes appearing in such a way are shown to be homeomorphic to balls or spheres. For pipe dream complexes, such strata correspond to slide polynomials.
Bibliography: 14 titles.
Keywords:
flag varieties, Schubert polynomials, Grothendieck polynomials, simplicial complexes.
The research of E. Yu. Smirnov was carried out with the support of the HSE University Basic Research Program, the Theoretical Physics and Mathematics Advancement Foundation “BASIS” (a ‘Junior Leader’ grant), the Russian Foundation for Basic Research (grant no. 20-01-00091-a), and the Simons Foundation (a Simons–IUM Fellowship). The research of A. A. Tutubalina was carried out with the support of the HSE University Basic Research Program and the Theoretical Physics and Mathematics Advancement Foundation “BASIS” (a ‘Junior Leader’ grant).