Big Varchenko-Gelfand rings and orbit harmonics

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.