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

2024/05/24 by Nobukazu Kowaki, Kowaki, Nobukazu
Mathematics · #13F65 #13P10 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Primary:14M15 #Secondary

paper · pdf · doi:10.48550/arxiv.2405.15215

openalex publication_date 2024/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citeclarke2021combinatorial. Our result can be regarded as a generalization of that result.

Related

• Combinatorial Mutations and Block Diagonal Polytopes
• Combinatorial Mutations of Gelfand-Tsetlin Polytopes, Feigin-Fourier-Littelmann-Vinberg Polytopes, and Block Diagonal Matching Field Polytopes
• Matching Fields in Macaulay2
• Geometry of tropical mutation surfaces with a single mutation
• Matching polytopes, toric geometry, and the non-negative part of the Grassmannian