Realization Spaces of Uniform Phased Matroids

TL;DR

This paper investigates the realization spaces of uniform phased matroids, revealing their simplicity compared to oriented matroids and providing a criterion for their realizability.

Contribution

It characterizes the realization spaces of uniform phased matroids and introduces a realizability criterion for those not essentially oriented.

Findings

Realization spaces of uniform phased matroids are simpler than those of uniform oriented matroids.

A criterion for realizability of uniform phased matroids is established.

Uniform phased matroids not essentially oriented have remarkably simple realization spaces.

Abstract

A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements. The realization space of an oriented (resp., phased) matroid is the space of vector arrangements in $\mathbb R^n$ (resp., $\mathbb C^n$) that correspond to oriented (resp., phased) matroid, modulo a change of coordinates. According to Mn\"ev's Universality Theorem, the realization spaces of uniform oriented matroids with rank greater than or equal to $3$ can be as complicated as any open semi-algebraic variety. In contrast, uniform phased matroids which are not essentially oriented have remarkably simple realization spaces if they are uniform. We also present a criterion for realizability of uniform phased matroids that are not essentially oriented.