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Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes

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arxiv 2411.11208 v1 pith:NTXJP2BG submitted 2024-11-17 math.CO math.AG

Allen Knutson , Paul Zinn-Justin This is my paper · ORCID
classification math.COmath.AG
keywords pipedreamsformulagenericclassesdoublelower-uppervarieties
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abstract

We recall the lower-upper varieties from [Knutson '05] and give a formula for their equivariant cohomology classes, as a sum over generic pipe dreams. We recover as limits the classic and bumpless pipe dream formulae for double Schubert polynomials. As a byproduct, we obtain a formula for the degree of the $n$th commuting variety as a sum of powers of 2. Generic pipe dreams also appear in the Segre-Schwarz-MacPherson analogue of the AJS/Billey formula, and when computing the Chern-Schwarz-MacPherson class of the orbit $B_- w B_+ \subseteq Mat_{k\times n}$ or of a double Bruhat cell $B_-u B_+ \cap B_+ v B_-$.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Flag positroid pipe dreams

    math.CO 2026-05 unverdicted novelty 7.0 of 10

    Flag positroid pipe dreams are new diagrams in bijection with Bruhat intervals whose elbow counts are Richardson-cell dimensions and whose row rule characterizes nonnegatively representable elementary positroid quotients.

  2. 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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