The Grothendieck Group of the Variety of Spanning Line Configurations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912791676321792 |
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| author | Zeng, Michael Ruofan |
| author_facet | Zeng, Michael Ruofan |
| contents | We study the Grothendieck group of the variety $X_{n,k}$ of spanning line configurations introduced by Pawlowski--Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra $R_{n,k}$. Our first result is a localization statement in $K$-theory for the complements of cell closures in smooth cellular varieties. Combining with the Fulton--Lascoux degeneracy loci formula, we prove that $K_0(X_{n,k})$ is canonically isomorphic to $R_{n,k}$, extending classical isomorphisms for the flag variety. We next identify the classes of the Pawlowski--Rhoades varieties with Grothendieck polynomials associated to words $w \in [k]^n$. Motivated by this identification, we develop models of classical and bumpless pipe dreams for words. We show that Schubert and Grothendieck polynomials of words are monomial-weight generating functions for these pipe dreams, extending the classical story from permutations to words and ordered set partitions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_22769 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Grothendieck Group of the Variety of Spanning Line Configurations Zeng, Michael Ruofan Combinatorics Algebraic Geometry 05E14 (Primary) 05E05, 19E08, 19E20 (Secondary) We study the Grothendieck group of the variety $X_{n,k}$ of spanning line configurations introduced by Pawlowski--Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra $R_{n,k}$. Our first result is a localization statement in $K$-theory for the complements of cell closures in smooth cellular varieties. Combining with the Fulton--Lascoux degeneracy loci formula, we prove that $K_0(X_{n,k})$ is canonically isomorphic to $R_{n,k}$, extending classical isomorphisms for the flag variety. We next identify the classes of the Pawlowski--Rhoades varieties with Grothendieck polynomials associated to words $w \in [k]^n$. Motivated by this identification, we develop models of classical and bumpless pipe dreams for words. We show that Schubert and Grothendieck polynomials of words are monomial-weight generating functions for these pipe dreams, extending the classical story from permutations to words and ordered set partitions. |
| title | The Grothendieck Group of the Variety of Spanning Line Configurations |
| topic | Combinatorics Algebraic Geometry 05E14 (Primary) 05E05, 19E08, 19E20 (Secondary) |
| url | https://arxiv.org/abs/2512.22769 |