Classification of external Zonotopal algebras

TL;DR

This paper classifies external Zonotopal algebras based on hyperplane arrangements, establishing a correspondence with zonotopes and regular matroids, and discusses related conjectures for central algebras.

Contribution

It provides a complete classification of external Zonotopal algebras and links unimodular cases to regular matroids, advancing the understanding of their algebraic structure.

Findings

External algebras classified by zonotopes

Unimodular external algebras correspond to regular matroids

Formulation of a conjecture for central algebras

Abstract

In this paper we work with power algebras associated to hyperplane arrangements. There are three main types of these algebras, namely, external, central, and internal zonotopal algebras. We classify all external algebras up to isomorphism in terms of zonotopes. Also, we prove that unimodular external zonotopal algebras are in one to one correspondence with regular matroids. For the case of central algebras we formulate a conjecture.