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

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.