Tropical cluster varieties of type C

TL;DR

This paper describes the tropicalization of a type C cluster variety as axially symmetric phylogenetic trees, analyzing sign patterns, dual polytopes, and constructing bases and degenerations.

Contribution

It provides an explicit description of the tropicalization of type C cluster varieties, linking them to phylogenetic trees and dual polytopes, with bases and degenerations.

Findings

Tropicalization realized as axially symmetric phylogenetic trees.

Sign patterns of coordinates are fully characterized.

Constructed Gr"obner and tropical bases, classified toric degenerations.

Abstract

We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and the cluster configuration space. We show that each of the corresponding signed tropicalizations is, combinatorially, dual to either a cyclohedron or an associahedron. As additional results, we construct Gr\"obner and tropical bases for the defining ideals of both varieties, and classify the arising toric degenerations.