Graded Big Varchenko–Gelfand Ring

• Graded big Varchenko–Gelfand ring is a filtered graded algebra built from oriented matroid data and hyperplane arrangements, unifying algebraic and topological insights.
• Its natural filtration, defined via Heaviside function degrees and indexed by flats, produces subquotients isomorphic to contracted VG rings with grading shifts.
• The ring features a canonical NBC basis with equivariant group actions, enabling applications in representation theory, cohomology, and combinatorial algebra.

The graded big Varchenko–Gelfand ring is a filtered graded algebra constructed from oriented matroid or hyperplane arrangement data. It encodes the algebraic and equivariant-geometric structure of locally constant functions on the real arrangement complement, generalizing the classical Varchenko–Gelfand ring to a setting that includes all covectors (faces) of a conditional oriented matroid. Its filtration and grading arise naturally from Heaviside function degrees and have deep ties to the topology of configuration spaces and equivariant cohomology.

1. Algebraic Definition and Filtration Structure

Let $\mathscr{M}$ denote a conditional oriented matroid with ground set $I$. The graded big Varchenko–Gelfand ring, denoted $\widehat{\mathscr{VG}_\mathscr{M}}$, is defined as the coordinate ring of the locus $Z_\mathscr{M}$ of covectors:

• Each covector $X$ yields a point in the affine space $\mathbb{k}^{I \times \{+, -, 0\}}$.
• The coordinate functions $(x)_{i,s}$ encode the position of $X$ at element $i$: $(x)_{i,s}=1$ iff $X(i)=s$, $s\in\{+, -, 0\}$.

Set $$\widehat{S} = \mathbb{k}[y_i^+, y_i^-, z_i : i \in I]$$ and define an ideal $I(Z_\mathscr{M})$ reflecting the combinatorics (e.g., $y_i^\pm(y_i^\pm - 1)$) and circuit relations.

• The coordinate ring $\widehat{S}/I(Z_\mathscr{M})$ is isomorphic to the ring of all functions $f$ on covectors (faces), with multiplication and addition given pointwise.
• The filtration is indexed by degree in Heaviside generators (the $y_i^\pm$ and $z_i$), i.e., $F_k$ is the span of all monomials of degree $\le k$.

The associated graded algebra is $$\widehat{\mathscr{VG}_\mathscr{M}} = \bigoplus_{k \geq 0} F_{k}/F_{k-1}$$, where $F_{-1} = 0$.

2. Filtration Indexed by Flats and Subquotient Decomposition

A distinguished algebra filtration is indexed by the poset $\mathscr{L}(\mathscr{M})$ of flats. For each flat $F$, select a minimal basic set $B(F) \subseteq I$ such that no proper subset generates $F$.

• Define $z_F = \prod_{b \in B(F)} z_b$.
• $z_{F_1}$ divides $z_{F_2}$ whenever $F_1 \subset F_2$.
• The filtered piece $$\widehat{\mathscr{VG}_\mathscr{M}}_F = z_F \cdot \widehat{\mathscr{VG}_\mathscr{M}}$$.

Subquotients indexed by flats are defined as $$\widehat{\mathscr{VG}_\mathscr{M}}_{=F} := \widehat{\mathscr{VG}_\mathscr{M}}_F / (\sum_{F' \supsetneq F} \widehat{\mathscr{VG}_\mathscr{M}}_{F'})$$.

• Theorem: These subquotients are (up to grading shift $-d_F$ with $d_F=|B(F)|$) isomorphic to the graded VG ring for the contraction $\mathscr{M}^F$:

$$\widehat{\mathscr{VG}_\mathscr{M}} \cong \bigoplus_{F \in \mathscr{L}(\mathscr{M})} \mathscr{VG}_{\mathscr{M}^F}(-d_F)$$

This divides the ring into direct summands corresponding to contractions at each flat.

3. No Broken Circuit Basis and Equivariant Structure

The canonical basis for $\widehat{\mathscr{VG}_\mathscr{M}}$ arises from the matroidal theory of no broken circuit (NBC) sets, crucial for both algebraic computation and representation theory.

• For each flat $F$, consider NBC sets $N$ of $\mathscr{M}^F$. The basis elements are:

$$m_{F,N} = \left( \prod_{b \in B(F)} z_b \right) \cdot \left( \prod_{i \in N} y_i^+ \right)$$

• The automorphism group $\mathrm{Aut}(\mathscr{M})$ acts naturally on all basis components via its action on $I$ and covector signs.
• The graded big VG ring decomposes as an $\mathrm{Aut}(\mathscr{M})$-module:

$$\widehat{\mathscr{VG}_\mathscr{M}} \cong \bigoplus_{[F] \in \mathscr{L}(\mathscr{M})/\mathrm{Aut}(\mathscr{M})} \mathrm{Ind}_{\mathrm{Aut}(\mathscr{M})_F}^{\mathrm{Aut}(\mathscr{M})} \left[ \mathscr{VG}_{\mathscr{M}^F}(-d_F) \right]$$

where induction is from the group stabilizer of $F$.

4. Orbit Harmonics Deformation

Orbit harmonics is the algebraic technique connecting the combinatorial data of covectors with graded ring structure by a flat limit deformation:

• Starting from the coordinate ring of the finite point set $Z_\mathscr{M}$, one replaces the defining ideal $I(Z_\mathscr{M})$ with its top homogeneous component (initial terms), producing a graded ring capturing "degree of vanishing" along $Z_\mathscr{M}$.
• This yields an isomorphism:

$$\widehat{\mathscr{VG}_\mathscr{M}} \cong \widehat{S} / (\tau(f) : f \in I(Z_\mathscr{M}), f \neq 0)$$

where $\tau(f)$ is the highest-degree part of $f$.

This deformation geometrizes the graded structure and enables analysis of Hilbert series and module decompositions, especially for arrangements with group symmetry (e.g., the braid arrangement case leads to Stirling-type Hilbert series distributions).

5. Topological Interpretation and Cohomological Connections

Equivariant cohomology provides the topological underpinning for the graded big Varchenko–Gelfand ring:

• For a hyperplane arrangement $\mathcal{A}$, the ring of locally constant functions (VG ring) is filtered via Heaviside degrees; its associated graded ring is the cohomology of the "3-arrangement" complement $M_3(\mathcal{A})$, and the full Rees algebra matches equivariant cohomology $H^\ast_T(M_3(\mathcal{A}))$ (Moseley, 2011, Dorpalen-Barry et al., 2022).
• The fixed point S$^1$-locus corresponds to the classical arrangement complement and the filtration specializes to the cohomological grading via the circle generator $u$.
• Similar geometric identifications extend to oriented matroids and their conditional analogs, yielding combinatorial and cohomological models for the graded big VG ring (Dorpalen-Barry et al., 2022).

6. Homological Algebra, Gelfand–Kirillov Dimension, and Canonical Modules

As a filtered-graded commutative ring, $\widehat{\mathscr{VG}_\mathscr{M}}$ inherits several homological and structural invariants:

• Its Gelfand–Kirillov dimension is computed via the growth rate of ranks of filtered pieces, and this value is preserved under passage to the associated graded ring (Lezama et al., 2019).
• In multigraded settings ($\mathbb{Z}^r$-graded), canonical module theory applies; localization and divisorial descriptions of the canonical module extend to affine monoid rings, suggesting analogous results for graded big VG rings (Barile et al., 16 May 2025).

7. Normality, Gelfand Graded Property, and Applications

The graded big Varchenko–Gelfand ring often possesses the Gelfand graded property: every homogeneous prime ideal sits in a unique graded maximal ideal, with associated Zariski retract and normality of the graded spectrum (Aqalmoun, 2022).

• In the subclass of pm$^+$ graded rings, the primes above any fixed homogeneous prime form a chain, yielding strong separation and duality properties crucial for homological applications and sheaf-theoretic analyses.

Applications of the graded big VG ring include:

• Representation theory (as a module for automorphism groups of arrangements).
• Topology (as a model for cohomology of configuration spaces and arrangement complements).
• Combinatorics (NBC bases, recursion on Poincaré polynomials, Orlik–Solomon type relations).
• Algebraic geometry (connection with Gröbner deformations, canonical module localization, and Abhyankar-type inequalities for graded multiplicities).

Summary Table: Core Features of the Graded Big Varchenko–Gelfand Ring

Feature	Description	Reference
Algebraic definition	Filtered graded ring of functions on covectors/faces	(Rhoades, 26 Aug 2025)
Filtration	Indexed by flats of the matroid, degree in Heaviside generators	(Rhoades, 26 Aug 2025)
NBC basis and group action	NBC sets basis compatible with automorphism group representation	(Rhoades, 26 Aug 2025)
Orbit harmonics deformation	Top homogeneous part yields graded ring from point locus	(Rhoades, 26 Aug 2025)
Cohomological/topological link	Associated graded matches cohomology of higher-dimensional complements	(Moseley, 2011, Dorpalen-Barry et al., 2022)
Homological invariants	GK-dimension, canonical module theory applies via multigrading	(Lezama et al., 2019, Barile et al., 16 May 2025)
Gelfand graded property	Prime spectrum normality, unique graded maximal ideals	(Aqalmoun, 2022)

The graded big Varchenko–Gelfand ring thus synthesizes combinatorial, algebraic, and topological theories, serving as a central object in the paper of arrangements, oriented matroids, and their associated invariants. Its structure is foundational for recent advances in equivariant cohomology, combinatorial commutative algebra, and representation theory.