Big Varchenko-Gelfand rings and orbit harmonics

2025/08/26 by Rhoades, Brendon
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.18602

Abstract

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

Citations

• Configuration spaces and peak representations
• Equivariant cohomology of Grassmannian spanning lines
• Increasing subsequences, matrix loci, and Viennot shadows
• Equivariant cohomology and conditional oriented matroids
• Packed Words and Quotient Rings
• Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture
• Enumeration of points, lines, planes, etc
• COMs: Complexes of Oriented Matroids
• The closure of a linear space in a product of lines
• The closure of a linear space in a product of lines
• Equivariant cohomology and the Varchenko-Gelfand filtration
• The equivariant Orlik-Solomon algebra

Related

• Reconstrucion of oriented matroids from Varchenko-Gelfand algebras
• Equivariant cohomology and conditional oriented matroids
• Graded Ehrhart theory and toric geometry
• Involution matrix loci and orbit harmonics
• A Broken Circuit Ring