Slide polynomials and subword complexes

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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.

About the authors

Evgeny Yurievich Smirnov

HSE University; Independent University of Moscow

Email: esmirnov@hse.ru
Candidate of physico-mathematical sciences, no status

Anna Alekseevna Tutubalina

HSE University

References

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  12. Е. Ю. Смирнов, А. А. Тутубалина, “Слайд-комплексы и комплексы подслов”, УМН, 75:6(456) (2020), 177–178
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