Realization spaces of matroids over hyperfields

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.