On Infinity Type Hyperplane Arrangements and Convex Positive Bijections

TL;DR

This paper proves that infinity type real hyperplane arrangements can be isomorphically represented with a convex positive bijection between their normal systems, establishing a precise condition for such representations.

Contribution

It establishes a necessary and sufficient condition for representing infinity type hyperplane arrangements via convex positive bijections between their normal systems.

Findings

Characterization of hyperplane arrangement isomorphisms

Equivalence of arrangements via convex positive bijections

Main theorem providing a criterion for arrangement representation

Abstract

In this article we prove in main Theorem A that any infinity type real hyperplane arrangement $\mathcal{H}_n^m$ (Definition 2.11) with the associated normal system $\mathcal{N}$ (Definitions [2.2,2.4] can be represented isomorphically (Definition 2.6) by another infinity type hyperplane arrangement $\tilde{\mathcal{H}}_n^m$ with a given associated normal system $\tilde{\mathcal{N}}$ if and only if the normal systems $\mathcal{N}$ and $\tilde{\mathcal{N}}$ are isomorphic, that is, there is a convex positive bijection (Definition 2.5) between a pair of associated sets of normal antipodal pairs of vectors of $\mathcal{N}$ and $\tilde{\mathcal{N}}$.