Varchenko–Gelfand Algebras in Hyperplane Arrangements

• Varchenko–Gelfand algebras are canonical commutative algebras defined as rings of locally constant functions on the complement of a hyperplane arrangement, reflecting its combinatorial and topological structure.
• They admit concrete presentations using Heaviside functions and orthogonal idempotents, which encode the chamber decomposition and intersection properties of the arrangement.
• These algebras are critical in topological analyses and computational reconstructions of oriented matroids, aiding in asphericity tests, Gröbner basis computations, and cohomology comparisons.

A Varchenko–Gelfand algebra (VG algebra) is a canonical commutative algebra associated to a real hyperplane arrangement or, more generally, to an oriented matroid. It arises as the ring of locally constant functions on the complement of the arrangement, with structure reflecting the combinatorics and topology of the underlying configuration. VG algebras admit concrete presentations in terms of Heaviside functions and satisfy a rich set of algebraic relations encoding intersection properties, circuit structure, and asphericity conditions of the arrangement and its complexification. They play a central role in the study of the topology of arrangement complements, reconstruction of oriented matroids, equivariant cohomology, and the broader theory of configuration spaces.

1. Definition and Classical Presentation

Let $\A = \{ H_1, \ldots, H_n \}$ be a finite, central, essential arrangement of hyperplanes in %%%%1%%%%. Denote the complement by $M_1(\A) = V \setminus \bigcup_{H \in \A} H$. The Varchenko–Gelfand algebra over a field $F$ is

$\VG(\A; F) = \{\,\text{locally constant functions }M_1(\A) \to F\,\},$

with pointwise addition and multiplication. Since $M_1(\A)$ decomposes as a disjoint union of open chambers (connected components), there is an abstract isomorphism $\VG(\A; F) \cong \bigoplus_{C \in \Ch(\A)} F$.

However, the core structure is given by a presentation in terms of Heaviside generators. For each $H \in \A$, choose a coorientation, producing half-spaces $H^+$ and $H^-$. Define $e_H^+$ and $e_H^-$ by:

$e_H^+(x) = \begin{cases} 1, & x \in H^+ \cap M_1(\A) \ 0, & x \in H^- \cap M_1(\A) \end{cases} \quad, \qquad e_H^- = 1 - e_H^+.$

These satisfy:

$$(e_H^\pm)^2 = e_H^\pm, \quad e_H^+e_H^- = 0, \quad e_H^+ + e_H^- = 1.$$

Letting $R = F[e_H^+, e_H^- \mid H \in \A] / \langle \text{idempotent relations as above} \rangle$, the canonical map

$\phi: R \to \VG(\A; F)$

is surjective. The kernel is generated by elements $g_S^\delta = f_S^\delta - f_S^{-\delta}$, where $f_S^\delta = \prod_{H \in S} e_H^{\delta_H}$ and $S \subseteq \A$, $\delta \in \{\pm\}^S$, with $\bigcap_{H \in S} H^{\delta_H} = \varnothing$; i.e., sign-vectors corresponding to empty intersections (Denham et al., 10 Dec 2025, Yagi et al., 24 Sep 2025).

The presentation is summarized as:

$\VG(\A; F) \cong F[e_H^\pm]/ \langle (e_H^\pm)^2 - e_H^\pm,\, e_H^+ + e_H^- - 1,\, e_H^+e_H^-,\, g_S^\delta \rangle.$

2. Algebraic Structure, Idempotent Basis, and Multiplication

Every full sign-vector $\epsilon \in \{\pm\}^\A$ specifies an element $f_\A^\epsilon = \prod_{H \in \A} e_H^{\epsilon_H}$. In the algebra $R$ these form $2^{|\A|}$ orthogonal idempotents; those corresponding to nonempty intersections survive in $\VG(\A; F)$ and label the chambers:

$f_\A^\epsilon(C) = \begin{cases} 1, & C \subseteq \bigcap_{H \in \A} H^{\epsilon_H} \ 0, & \text{otherwise}. \end{cases}$

Thus, the set $\{f_\A^\epsilon \mid \epsilon \in \Ch(\A) \}$ provides an $F$-basis. The multiplication law is:

$f_\A^\epsilon \cdot f_\A^\eta = \delta_{\epsilon, \eta} f_\A^\epsilon.$

This orthogonal idempotent structure encodes the chamber decomposition and supports explicit computation (Denham et al., 10 Dec 2025).

3. Filtrations, Graded Structure, and Orlik–Solomon Connections

The VG algebra is naturally filtered by degree with respect to Heaviside generators:

$$F^d = \text{span}\{\, \text{monomials in the }e_H^+ \text{ of degree} \le d\,\}.$$

The associated graded algebra $\operatorname{gr} \VG(\A; F) = \bigoplus_{d \ge 0} F^d/F^{d-1}$ interprets the leading monomials, with relations reflecting underlying matroid circuits. When the coefficient ring has characteristic $2$, $\operatorname{gr} \VG$ is isomorphic to the Orlik–Solomon algebra of the arrangement, matching the cohomology ring of the complement (Yagi et al., 24 Sep 2025, Moseley, 2011). For other characteristics, sign data in circuit relations modifies the isomorphism, and full correspondence occurs only after reduction modulo $2$.

For oriented matroids, the construction extends: the VG algebra is the ring of functions on the set of chambers/topes, with generators and relations determined by signed circuits (Moseley, 2011).

4. Combinatorial Criteria and Yoshinaga’s Theorem

A central problem concerns asphericity: determining when the complexification $M_2(\A) = (V \otimes_\R \C) \setminus \bigcup_{H \in \A} (H \otimes_\R \C)$ is a $K(\pi, 1)$. Yoshinaga provides a combinatorial criterion:

• Define $k$-consistent sign-vectors $\epsilon \in \{\pm\}^\A$ as those for which every subset $S \subseteq \A$, $|S| \leq k+1$, satisfies $\bigcap_{H \in S} H^{\epsilon_H} \neq \varnothing$. Set $\sigma_k(\A)$ to be the number of such vectors.
• Yoshinaga's criterion: $\A$ satisfies it when $\sigma_2(\A) = \sigma_r(\A) = |\Ch(\A)|$, where $r = \text{rank}(\A)$.

Denham–Dorpalen-Barry–Proudfoot demonstrate that this combinatorial criterion is equivalent to the algebraic condition $I_2 = I_r$ for the ideal $I_k$ generated by the degree $\leq k+1$ relations $g_S^\delta$ as above. In effect, $\VG_2(\A; F) \cong \VG_r(\A; F)$ exactly when Yoshinaga's asphericity test is satisfied. This equivalence enables Gröbner basis computation of asphericity obstructions, significantly improving efficiency compared to direct enumeration (Denham et al., 10 Dec 2025).

5. Matroid and Oriented Matroid Reconstruction

For arrangements generic in codimension $2$, both the filtered and graded VG algebras determine the underlying oriented matroid up to isomorphism. Explicit algorithms:

• For the filtered VG algebra, enumerate all bases for degree-$1$ idempotents, build the (generalized) tope graph, and check for matroid structure.
• For the associated graded, enumerate generating tuples of square-zero elements, identify minimal vanishing product subsets (matroid circuits), and reconstruct signed-circuit data.

For generic codimension-$2$ cases, isomorphism of filtered or graded VG algebras implies isomorphism of oriented matroids. For non-generic arrangements, a conjectural algorithm explores all possible degree-$1$ bases and verifies whether the resulting structure matches an oriented matroid (Yagi et al., 24 Sep 2025).

6. Big Varchenko–Gelfand Algebras and Orbit Harmonics

Rhoades introduces "big" VG algebras $\widehat{VG}_M$ for conditional oriented matroids $M$, which incorporate covector data beyond chambers. These graded algebras $\widehat{VG}_M$ are constructed via generators $y_i^\pm, z_i$ subject to idempotent, sum-to-zero, and circuit relations; their subquotients (indexed by flats of the matroid) recover the "small" VG algebras of contractions $M^F$, shifted in grading. The big VG algebra thus decomposes as a sum over the classical VG rings of all contractions:

$$\widehat{VG}_M \cong \bigoplus_{F \in L(M)} VG_{M^F}(-\operatorname{rk}(F)).$$

A distinguished no–broken–circuit basis parametrizes $\widehat{VG}_M$. The construction involves orbit harmonics, connecting the algebra to the flat limit of the vanishing ideal describing the covector locus $Z_M \subset \Bbbk^{I \times \{+, -, 0\}}$. The equivariant structure under the automorphism group of $M$ is explicit, and the big VG algebra admits a natural grading and flat-indexed filtration (Rhoades, 26 Aug 2025).

7. Topological and Equivariant Cohomological Context

The VG filtration has an interpretation in terms of equivariant cohomology. For the configuration space $M_3(\A)$ (the space of triples in $V^3$ satisfying arrangement-induced nonvanishing conditions), the associated graded of the VG algebra recovers $H^*(M_3(\A))$ as a graded ring. Thus, the ring structure aligns with topological invariants arising from the configuration space, and, in the case of the braid arrangement, the cohomology admits a realization as the regular representation of the symmetric group. Extension to affine arrangements and oriented matroids proceeds via combinatorial presentations analogous to those for classical arrangements (Moseley, 2011).

8. Computational Methods and Practical Relevance

The algebraic description of VG algebras enables effective computational techniques for arrangement problems, particularly for checking asphericity (Yoshinaga's criterion) and reconstructing matroidal data. Computations involving Gröbner bases in polynomial rings with idempotent and circuit relations are tractable for sizable examples, outperforming brute-force combinatorial enumeration. VG algebras thus play a central, practical role in the algebraic and topological analysis of arrangements (Denham et al., 10 Dec 2025).