Big Varchenko-Gelfand rings and orbit harmonics
Brendon Rhoades
TL;DR
The paper defines a graded big Varchenko-Gelfand ring $\widehat{\mathcal{VG}}_\mathcal{M}$ attached to a conditional oriented matroid $\mathcal{M}$, and shows it carries a filtration by the flats poset $\mathcal{L}(\mathcal{M})$ whose subquotients are the small VG rings of contractions $\mathcal{M}^F$. It provides an explicit presentation and generators for the big locus $\mathcal{Z}_\mathcal{M}$, proves a precise isomorphism $\mathbf{R}(\mathcal{Z}_\mathcal{M}) \cong \widehat{\mathcal{VG}}_\mathcal{M}$ via orbit harmonics, and establishes a decomposition into contraction contributions $\widehat{\mathcal{VG}}_\mathcal{M} \cong \bigoplus_{F\in\mathcal{L}(\mathcal{M})} \mathcal{VG}_{\mathcal{M}^F}(-\mathrm{codim}(F))$, with an enhanced equivariant structure. The braid arrangement is worked out as a key example, yielding explicit Hilbert series and a detailed $\mathfrak{S}_n$-module description that connects to configuration-space cohomology and Eulerian representations; the paper also raises a topological interpretation problem for the big VG ring. Overall, it extends the orbit-harmonics framework to a big-locus setting, revealing how big and small VG rings interrelate via flats and contractions and enabling representation-theoretic commentary in combinatorial geometry. The results provide new algebraic tools for studying covector and tope structures in oriented matroids and their symmetry actions.
Abstract
Let $\mathscr{M}$ be a conditional oriented matroid. We define a graded algebra $\widehat{\mathscr{VG}}_\mathscr{M}$ with vector space dimension given by the number of covectors in $\mathscr{M}$ which admits a distinguished filtration indexed by the poset $\mathscr{L}(\mathscr{M})$ of flats of $\mathscr{M}$. The subquotients of this filtration are isomorphic to graded Varchenko-Gelfand rings of contractions of $\mathscr{M}$, so we call $\widehat{\mathscr{VG}}_\mathscr{M}$ the {\em graded big Varchenko-Gelfand ring of $\mathscr{M}$.} We describe a no broken circuit type basis of $\widehat{\mathscr{VG}}_\mathscr{M}$ and study its equivariant structure under the action of $\mathrm{Aut}(\mathscr{M})$. Our key technique is the orbit harmonics deformation which encodes $\widehat{\mathscr{VG}}_\mathscr{M}$ (as well as the classical Varchenko-Gelfand ring) in terms of a locus of points.