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arXiv · 2509.19905

Reconstruction of oriented matroids from Varchenko-Gelfand algebras

Yukino Yagi·Masahiko Yoshinaga
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Abstract

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.

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Yukino Yagi, Masahiko Yoshinaga. 2026-05-26. Reconstruction of oriented matroids from Varchenko-Gelfand algebras. https://arxiv.org/abs/2509.19905

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