Reconstrucion of oriented matroids from Varchenko-Gelfand algebras

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.

💡 Research Summary

The paper investigates Varchenko‑Gelfand (VG) algebras associated with real central hyperplane arrangements, focusing on coefficient rings R that are integral domains of characteristic different from 2. The VG algebra VG(A) consists of all R‑valued functions on the set of chambers of an arrangement A, and it is generated by Heaviside functions x_i^{±} corresponding to the two half‑spaces of each hyperplane H_i. These generators induce a natural filtration F_k by polynomial degree, and the associated graded algebra VG·(A) is obtained as F_k/F_{k-1}. When char R = 2, VG·(A) coincides with the Orlik‑Solomon algebra, but for char R ≠ 2 the situation is largely unexplored.

The authors’ main contribution is to show that, under a modest genericity hypothesis—namely that the arrangement is “generic in codimension 2” (any three distinct hyperplanes intersect in codimension 3)—the VG algebra determines the underlying oriented matroid. Two complementary reconstruction results are proved:

1. Filtered reconstruction (Theorem 3.6).
   Primitive idempotents of VG(A) are exactly the characteristic functions 1_C of chambers C. The degree‑one part F_1 contains precisely the Heaviside functions, which can be identified as the non‑trivial idempotents in F_1. By extracting these idempotents one recovers the set of hyperplanes, their orientations, and the adjacency relation between chambers (the “top‑graph” T(A)). Consequently, the filtered VG algebra F_⋅VG(A) encodes the full top‑graph, and therefore the oriented matroid, because an oriented matroid is uniquely determined by its top‑graph (Proposition 2.1).

2. Graded reconstruction (Theorem 5.3).
   In the graded algebra VG·(A), degree‑one elements whose square is zero (square‑zero elements) correspond to signed circuits of the oriented matroid. The authors show that each signed circuit σ gives rise to a unique linear relation among monomials of degree k‑1 in the generators, mirroring the circuit relations in the Orlik‑Solomon algebra. By analyzing these relations one can recover the entire set of signed circuits C(A). Since the circuit set determines the oriented matroid, the graded VG algebra also suffices for reconstruction.

The paper also demonstrates that several classical invariants—intersection lattice L(A), Orlik‑Solomon algebra, and Poincaré polynomial—do not determine the VG algebra, providing explicit counterexamples (Examples 3.11, 5.2, 5.4). This underscores that VG algebras contain strictly more combinatorial information than these traditional invariants.

Beyond the two theorems, the authors propose a conjectural, non‑deterministic algorithm (Section 4) for reconstructing the top‑graph from a filtered VG algebra in the general (non‑generic) case. The algorithm relies on identifying all idempotents in F_1, interpreting them as “generalized Heaviside functions,” and iteratively building the adjacency structure. Its termination is linked to a Sylvester‑Gallai type condition, and the authors conjecture that it always succeeds for any real arrangement.

A notable side result concerns “3‑plexifications” (real arrangements tensored with ℝ³). The authors prove that, unlike complexifications, the cohomology ring of the complement of a 3‑plexification is not determined by the intersection lattice, highlighting a new phenomenon in real‑to‑complex extensions.

In summary, the paper establishes that for a wide class of real arrangements (generic in codimension 2) the VG algebra—both filtered and graded—encodes the full oriented matroid structure, even over coefficient rings of odd characteristic. It clarifies the precise relationship between VG algebras, oriented matroids, and classical combinatorial invariants, and it opens a promising line of research on algorithmic reconstruction of oriented matroids from algebraic data.