The geometry of zonotopal algebras I: cohomology of graphical configuration spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913830952501248 |
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| author | Crowley, Colin Dorpalen-Barry, Galen Henriques, André Proudfoot, Nicholas |
| author_facet | Crowley, Colin Dorpalen-Barry, Galen Henriques, André Proudfoot, Nicholas |
| contents | Zonotopal algebras of vector arrangements are combinatorially-defined algebras with connections to approximation theory, introduced by Holtz and Ron and independently by Ardila and Postnikov. We show that the internal zonotopal algebra of a cographical vector arrangement is isomorphic to the cohomology ring of a certain configuration space introduced by Moseley, Proudfoot, and Young. We also study an integral form of this algebra, which in the cographical case is isomorphic to the integral cohomology ring. Our results rely on interpreting the internal zonotopal algebra of a totally unimodular arrangement as an orbit harmonics ring, that is, as the associated graded of the ring of functions on a finite set of lattice points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12768 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The geometry of zonotopal algebras I: cohomology of graphical configuration spaces Crowley, Colin Dorpalen-Barry, Galen Henriques, André Proudfoot, Nicholas Combinatorics 05B35, 05C50, 52B40, 52C07, 52C35 Zonotopal algebras of vector arrangements are combinatorially-defined algebras with connections to approximation theory, introduced by Holtz and Ron and independently by Ardila and Postnikov. We show that the internal zonotopal algebra of a cographical vector arrangement is isomorphic to the cohomology ring of a certain configuration space introduced by Moseley, Proudfoot, and Young. We also study an integral form of this algebra, which in the cographical case is isomorphic to the integral cohomology ring. Our results rely on interpreting the internal zonotopal algebra of a totally unimodular arrangement as an orbit harmonics ring, that is, as the associated graded of the ring of functions on a finite set of lattice points. |
| title | The geometry of zonotopal algebras I: cohomology of graphical configuration spaces |
| topic | Combinatorics 05B35, 05C50, 52B40, 52C07, 52C35 |
| url | https://arxiv.org/abs/2502.12768 |