Oriented Matroids from Triangulations of Products of Simplices

Marcel Celaya, Georg Loho, Chi Ho Yuen

Published 2020-05-04Version 1

We introduce a construction of oriented matroids from a triangulation of a product of two simplices. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. In particular, we generalize this using the language of matroids over hyperfields, which gives a new approach to construct matroids over hyperfields. Using the polyhedral structure, we derive a topological representation of the oriented matroid. This relies on a variant of Viro's patchworking and insights on the structure of tropical oriented matroids. A recurring theme in our work is that various tropical constructions can be extended beyond tropicalization with new formulations and proof methods.