Hyperfield Grassmannians

TL;DR

This paper extends the concept of Grassmannians to topological hyperfields, connecting matroid theory, tropical geometry, and classical algebraic structures, and explores their realization spaces.

Contribution

It introduces topological hyperfields and generalizes Grassmannians and realization spaces within this new framework, unifying various mathematical theories.

Findings

Established a framework for topological hyperfields

Generalized Grassmannians to hyperfield context

Linked hyperfields in matroid theory and tropical geometry

Abstract

In a recent paper Baker and Bowler introduced matroids over hyperfields, offering a common generalization of matroids, oriented matroids, and linear subspaces of based vector spaces. This paper introduces the notion of a topological hyperfield and explores the generalization of Grassmannians and realization spaces to this context, particularly in relating the (hyper)fields R and C to hyperfields arising in matroid theory and in tropical geometry.