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Tropical cluster varieties, phylogenetic trees, and generalized associahedra

Igor Makhlin

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.

Tropical cluster varieties, phylogenetic trees, and generalized associahedra

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.
Paper Structure (3 sections, 8 theorems, 4 equations, 4 figures)

This paper contains 3 sections, 8 theorems, 4 equations, 4 figures.

Table of Contents

  1. ASPTs and the space of ASPTs
  2. Type C cluster algebras and the main theorem
  3. Signed tropicalizations and generalized associahedra

Key Result

Theorem A

For a full-rank cluster variety $X(\tilde{B})$ of finite type $\mathrm C_{n-1}$, the tropical cluster variety$\mathop{\mathrm{Trop}}\nolimits X(\tilde{B})$ is, modulo lineality space, equal to the space of ASPTs.

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure 3: For $n=3$, the space of ASPTs is, modulo lineality space, a fan over this graph. Some elements are labeled by the respective ASPTs.
  • Figure 4:

Theorems & Definitions (22)

  • Theorem A: Thm. \ref{['thm:main']}, Cor. \ref{['cor:fullrank']}
  • Theorem B: Thm. \ref{['thm:signedtrops']}
  • Example 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.5
  • Definition 1.6
  • Proposition 1.7
  • Example 1.8
  • ...and 12 more