Séminaire Lotharingien de Combinatoire, 93B.27 (2025), 12 pp.

Allen Knutson and Paul Zinn-Justin

Generic Pipe Dreams, Lower-Upper Varieties, and Schwartz-MacPherson Classes

Abstract. We recall the lower-upper varieties E_w 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 nth commuting variety as a sum of powers of 2.

Generic pipe dreams also appear in the Segre-Schwartz-MacPherson analogue of the AJS/Billey formula, and when computing the Chern-Schwartz-MacPherson class of the orbit B_-wB₊ ⊆ Mat_{k × n} or of a double Bruhat cell B_-uB₊ ∩ B₊vB_-.

Received: November 15, 2024. Accepted: February 15, 2025. Final version: April 1, 2025.

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