Big Varchenko-Gelfand rings and orbit harmonics

Abstract

Let $\mathcal{M}$ be a conditional oriented matroid. We define a graded algebra ${\widehat{\mathcal{VG}}}_{\mathcal{M}}$ with vector space dimension given by the number of covectors in $\mathcal{M}$ which admits a distinguished filtration indexed by the poset $\mathcal{L}(\mathcal{M})$ of flats of $\mathcal{M}$. The subquotients of this filtration are isomorphic to graded Varchenko-Gelfand rings of contractions of $\mathcal{M}$, so we call ${\widehat{\mathcal{VG}}}_{\mathcal{M}}$ the {\em graded big Varchenko-Gelfand ring of $\mathcal{M}$.} We describe a no broken circuit type basis of ${\widehat{\mathcal{VG}}}_{\mathcal{M}}$ and study its equivariant structure under the action of $\mathrm{Aut}(\mathcal{M})$. Our key technique is the orbit harmonics deformation which encodes ${\widehat{\mathcal{VG}}}_{\mathcal{M}}$ (as well as the classical Varchenko-Gelfand ring) in terms of a locus of points.

26 pages. Prop. 5.1 and Example 5.2 added in this version