Tropicalizing binary geometries
Shelby Cox, Igor Makhlin
TL;DR
The paper develops a unified framework to tropicalize binary geometries arising from type A and type C cluster algebras, recasting the tropicalization of $\mathcal{M}_{0,n}$ as the space of phylogenetic trees and extending this picture to the cyclohedron via axially and centrally symmetric phylogenetic trees. It introduces the complexes $\Theta_{\mathrm{as}}(n)$ and $\Theta_{\mathrm{cs}}(n)$ to model the tropical structure, and conjectures that $\mathrm{Trop}\mathcal{M}_{\mathrm{C}_{n-1}}$ is the fan over the ASPT complex, with signed tropicalizations corresponding to dual associahedra and dual cyclohedra subfans. The work provides explicit $u$-coordinate descriptions for $\mathcal{M}_{0,n}$ and outlines a concrete combinatorial realization for type C tropical geometry, connecting to type C tropicalized cluster varieties. The $n=3$ case $\mathrm{Trop}\mathcal{M}_{\mathrm{C}_2}$ is validated computationally, supporting the proposed framework and offering potential avenues for understanding type C compactifications and their stratifications.
Abstract
The type A cluster configuration space, commonly known as $\mathcal M_{0,n}$, is the very affine part of the binary geometry associated with the associahedron. The tropicalization of $\mathcal M_{0,n}$ can be realized as the space of phylogenetic trees and its signed tropicalizations as the dual-associahedron subfans. We give a concise overview of this construction and propose an extension to type C. The type C cluster configuration space $\mathcal M_{\mathrm C_l}$ arises from the binary geometry associated with the cyclohedron. We define a space of axially symmetric phylogenetic trees containing many dual-associahedron and dual-cyclohedron subfans. We conjecturally realize the tropicalization of $\mathcal M_{\mathrm C_l}$ as the defined space and its signed tropicalizations as the aforementioned subfans.