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Schubert polynomials, pipe dreams, equivariant classes, and a co-transition formula

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arxiv 1909.13777 v2 pith:JOBV43YZ submitted 2019-09-30 math.CO math.AG

Allen Knutson This is my paper · ORCID
classification math.COmath.AG
keywords polynomialsformulaschubertclassesco-transitionequivariantfamilieslascoux
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We give a new proof that three families of polynomials coincide: the double Schubert polynomials of Lascoux and Sch\"utzenberger defined by divided difference operators, the pipe dream polynomials of Bergeron and Billey, and the equivariant cohomology classes of matrix Schubert varieties. All three families are shown to satisfy a "co-transition formula" which we explain to be some extent projectively dual to Lascoux' transition formula. We comment on the K-theoretic extensions.

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  1. Hybrid pipe dreams for the lower-upper scheme

    math.CO 2025-09 conditional novelty 7.0 of 10

    Hybrid generic pipe dreams give a hybridization-independent equivariant formula for the classes of lower-upper varieties, proved via Yang-Baxter equations and a degeneration into complete intersections.

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