The geometry of zonotopal algebras I: cohomology of graphical configuration spaces

TL;DR

This paper establishes a deep connection between zonotopal algebras of cographical arrangements and the cohomology rings of specific configuration spaces, revealing new algebraic-topological relationships.

Contribution

It proves that the internal zonotopal algebra of cographical arrangements is isomorphic to the cohomology ring of certain configuration spaces, linking combinatorics, algebra, and topology.

Findings

Isomorphism between internal zonotopal algebra and cohomology ring

Integral form of algebra matches integral cohomology

Interpretation as orbit harmonics ring

Abstract

Zonotopal algebras of vector arrangements are combinatorially-defined algebras with connections to approximation theory, introduced by Holtz and Ron and independently by Ardila and Postnikov. We show that the internal zonotopal algebra of a cographical vector arrangement is isomorphic to the cohomology ring of a certain configuration space introduced by Moseley, Proudfoot, and Young. We also study an integral form of this algebra, which in the cographical case is isomorphic to the integral cohomology ring. Our results rely on interpreting the internal zonotopal algebra of a totally unimodular arrangement as an orbit harmonics ring, that is, as the associated graded of the ring of functions on a finite set of lattice points.