{"paper":{"title":"Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Allen Knutson, Paul Zinn-Justin","submitted_at":"2024-11-17T23:47:58Z","abstract_excerpt":"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_+ \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.11208","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.11208/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}