Real phase structures on matroid fans and matroid orientations

TL;DR

This paper introduces real phase structures on rational polyhedral fans, establishing a correspondence with matroid orientations, and provides methods to recover matroid properties from these structures.

Contribution

It defines real phase structures on fans and proves their equivalence to matroid orientations, offering new tools to analyze matroid properties.

Findings

Real phase structures are equivalent to matroid orientations.

Topes of the oriented matroid are directly recoverable from the structure.

Signed circuits can be reconstructed from the real phase structure.

Abstract

We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two conditions. Given an oriented matroid we can construct a real phase structure on the fan of the underlying matroid. Conversely, we show that from a real phase structure on a matroid fan we can produce an orientation of the underlying matroid. Thus real phase structures are cryptomorphic to matroid orientations. The topes of the orientated matroid are recovered immediately from the real phase structure. We also provide a direct way to recover the signed circuits of the oriented matroid from the real phase structure.