Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913579894046720 |
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| author | Knutson, Allen Zinn-Justin, Paul |
| author_facet | Knutson, Allen Zinn-Justin, Paul |
| contents | 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 $B_-u B_+ \cap B_+ v B_-$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes Knutson, Allen Zinn-Justin, Paul Combinatorics Algebraic Geometry 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 $B_-u B_+ \cap B_+ v B_-$. |
| title | Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes |
| topic | Combinatorics Algebraic Geometry |
| url | https://arxiv.org/abs/2411.11208 |