Big Varchenko-Gelfand rings and orbit harmonics

TL;DR

This paper introduces a new graded algebra called the big Varchenko-Gelfand ring associated with conditional oriented matroids, exploring its structure, bases, and symmetries through orbit harmonics deformation.

Contribution

It defines the graded big Varchenko-Gelfand ring for conditional oriented matroids and analyzes its structure, bases, and equivariant properties using orbit harmonics deformation.

Findings

Established a basis of no broken circuit type for the algebra.

Described the algebra's equivariant structure under automorphisms.

Connected the algebra to a locus of points via orbit harmonics deformation.

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.