Tropical hyperplane arrangements and combinatorial mutations of the matching field polytopes of Grassmannians

TL;DR

This paper introduces a method using tropical hyperplane arrangements to quickly determine combinatorial mutation equivalence of matching field polytopes, aiding in understanding toric degenerations of Grassmannians.

Contribution

It provides a new, efficient way to verify combinatorial mutation equivalence of matching field polytopes via tropical hyperplane arrangements, generalizing previous results.

Findings

Block diagonal matching fields are mutation equivalent to diagonal matching fields.

The method simplifies checking mutation equivalence at a glance.

Generalizes previous results on matching field polytopes.

Abstract

A sequence of combinatorial mutations of matching field polytopes preserves the property of giving rise to a toric degeneration of Grassmannians. In this paper, we find a way to check that two matching field polytopes are combinatorial mutation equivalence using tropical hyperplane arrangements, ``literally at a glance". Our way can prove that block diagonal matching fields are combinatorial mutations equivalent to diagonal matching fields. This is one of main results in \cite{clarke2021combinatorial}. Our result can be regarded as a generalization of that result.