Tropical cluster varieties, phylogenetic trees, and generalized associahedra

TL;DR

This paper describes the tropicalization of a type C cluster variety as a space of symmetric phylogenetic trees and explores its signed tropicalizations related to associahedra and cyclohedra.

Contribution

It explicitly characterizes the tropicalization of a type C cluster variety and links it to phylogenetic trees and generalized associahedra.

Findings

Tropicalization corresponds to axially symmetric phylogenetic trees.

Signed tropicalizations are subfans dual to associahedra or cyclohedra.

Provides explicit geometric descriptions of these tropicalizations.

Abstract

We explicitly describe the tropicalization of a type C cluster variety by identifying it with the space of axially symmetric phylogenetic trees. We also study the signed tropicalizations of this cluster variety, realizing them as subfans of the tropicalization that are dual to either associahedra or cyclohedra.