Reconstruction of oriented matroids from Varchenko-Gelfand algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910259339067392 |
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| author | Yagi, Yukino Yoshinaga, Masahiko |
| author_facet | Yagi, Yukino Yoshinaga, Masahiko |
| contents | The algebra of $R$-valued functions on the set of chambers of a real hyperplane arrangement is called the Varchenko-Gelfand (VG) algebra. This algebra carries a natural filtration by the degree with respect to Heaviside functions, giving rise to the associated graded VG algebra. When the coefficient ring $R$ is an integral domain of characteristic $2$, the graded VG algebra is known to be isomorphic to the Orlik-Solomon algebra. In this paper, we study VG algebras over coefficient rings of characteristic different from $2$, and investigate to what extent VG algebras determine the underlying oriented matroid structures.
Our main results concern hyperplane arrangements that are generic in codimension $2$. For such arrangements, if $R$ is an integral domain of characteristic not equal to $2$, then the oriented matroid can be recovered from both the filtered and the graded VG algebras. As a byproduct, we prove that, unlike the complexification, the cohomology ring of the complement of a $3$-plexification of a real arrangement is not determined by the intersection lattice.
We also formulate an algorithm that is expected to reconstruct oriented matroids from VG algebras in the case of general arrangements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_19905 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reconstruction of oriented matroids from Varchenko-Gelfand algebras Yagi, Yukino Yoshinaga, Masahiko Combinatorics Commutative Algebra Algebraic Geometry 52C35, 51M20, 32S22, 05B35, 52C40 The algebra of $R$-valued functions on the set of chambers of a real hyperplane arrangement is called the Varchenko-Gelfand (VG) algebra. This algebra carries a natural filtration by the degree with respect to Heaviside functions, giving rise to the associated graded VG algebra. When the coefficient ring $R$ is an integral domain of characteristic $2$, the graded VG algebra is known to be isomorphic to the Orlik-Solomon algebra. In this paper, we study VG algebras over coefficient rings of characteristic different from $2$, and investigate to what extent VG algebras determine the underlying oriented matroid structures. Our main results concern hyperplane arrangements that are generic in codimension $2$. For such arrangements, if $R$ is an integral domain of characteristic not equal to $2$, then the oriented matroid can be recovered from both the filtered and the graded VG algebras. As a byproduct, we prove that, unlike the complexification, the cohomology ring of the complement of a $3$-plexification of a real arrangement is not determined by the intersection lattice. We also formulate an algorithm that is expected to reconstruct oriented matroids from VG algebras in the case of general arrangements. |
| title | Reconstruction of oriented matroids from Varchenko-Gelfand algebras |
| topic | Combinatorics Commutative Algebra Algebraic Geometry 52C35, 51M20, 32S22, 05B35, 52C40 |
| url | https://arxiv.org/abs/2509.19905 |