Oriented Matroids from Triangulations of Products of Simplices
Classically, there is a rich theory in algebraic combinatorics surrounding the various objects associated with a generic real matrix. Examples include regular triangulations of the product of two simplices, coherent matching fields, and realizable oriented matroids. In this talk, we will extend the theory by skipping the matrix and starting with an arbitrary triangulation of the product of two simplices instead. In particular, we show that every polyhedral matching field induces
oriented matroids. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex. Furthermore, we give a corresponding topological construction using Viro's patchworking. This talk will also sketch the relationship between Baker-Bowler's matroids over hyperfields and our work. This is joint work with Marcel Celaya and Chi-Ho Yuen.