Realization spaces of matroids over hyperfields

TL;DR

This paper explores the structure and properties of realization spaces of matroids over hyperfields, providing new descriptions, topological insights, and applications to phased matroids and hyperplane arrangements.

Contribution

It introduces explicit models for realization spaces over hyperfields, linking them to hyperfield Grassmannians and providing tools for computation and topological analysis.

Findings

Realization spaces can be described via Tutte groups and projective classes.

Topological hyperfield realization spaces have the correct homeomorphism type.

Applications include non-realizability of certain phased matroids and insights into hyperplane arrangement topology.

Abstract

We study realization spaces of matroids over hyperfields (in the sense of Baker and Bowler). More precisely, given a matroid M and a hyperfield H we determine the space of all H-matroids over M. This can be seen as the matroid stratum of the hyperfield Grassmannians in the sense of Anderson and Davis. We give different descriptions of these realization spaces (e.g., in terms of Tutte groups or projective classes), allowing for explicit computations. When the hyperfield at hand is topological, the realization spaces have a natural topology. In this case, our models carry the correct homeomorphism type. As applications of our methods we obtain a theorem on the existence of phased matroids that are not realizable over the complex field and are not chirotopal, as well as a result on the diffeomorphism type of complex hyperplane arrangements whose underlying matroid is uniform.