The geometry of zonotopal algebras II: Orlik--Terao algebras and Schubert varieties
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909604676370432 |
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| author | Crowley, Colin Proudfoot, Nicholas |
| author_facet | Crowley, Colin Proudfoot, Nicholas |
| contents | Zonotopal algebras, introduced by Postnikov--Shapiro--Shapiro, Ardila--Postnikov, and Holtz--Ron, show up in many different contexts, including approximation theory, representation theory, Donaldson--Thomas theory, and hypertoric geometry. In the first half of this paper, we construct a perfect pairing between the internal zonotopal algebra of a linear space and the reduced Orlik--Terao algebra of the Gale dual linear space. As an application, we prove a conjecture of Moseley--Proudfoot--Young that relates the reduced Orlik--Terao algebra of a graph to the cohomology of a certain configuration space. In the second half of the paper, we interpret the Macaulay inverse system of a zonotopal algebra as the space of sections of a sheaf on the Schubert variety of a linear space. As an application of this, we use an equivariant resolution of the structure sheaf of the Schubert variety inside of a product of projective lines to produce an exact sequence relating internal and external zonotopal algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The geometry of zonotopal algebras II: Orlik--Terao algebras and Schubert varieties Crowley, Colin Proudfoot, Nicholas Algebraic Geometry Combinatorics Zonotopal algebras, introduced by Postnikov--Shapiro--Shapiro, Ardila--Postnikov, and Holtz--Ron, show up in many different contexts, including approximation theory, representation theory, Donaldson--Thomas theory, and hypertoric geometry. In the first half of this paper, we construct a perfect pairing between the internal zonotopal algebra of a linear space and the reduced Orlik--Terao algebra of the Gale dual linear space. As an application, we prove a conjecture of Moseley--Proudfoot--Young that relates the reduced Orlik--Terao algebra of a graph to the cohomology of a certain configuration space. In the second half of the paper, we interpret the Macaulay inverse system of a zonotopal algebra as the space of sections of a sheaf on the Schubert variety of a linear space. As an application of this, we use an equivariant resolution of the structure sheaf of the Schubert variety inside of a product of projective lines to produce an exact sequence relating internal and external zonotopal algebras. |
| title | The geometry of zonotopal algebras II: Orlik--Terao algebras and Schubert varieties |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2505.05324 |