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Scattering diagrams for generalized cluster algebras

Lang Mou

Abstract

We construct scattering diagrams for Chekhov-Shapiro's generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov's cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors.

Scattering diagrams for generalized cluster algebras

Abstract

We construct scattering diagrams for Chekhov-Shapiro's generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov's cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors.

Paper Structure

This paper contains 45 sections, 49 theorems, 260 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized cluster algebras
  3. Generalized cluster varieties
  4. Scattering diagrams
  5. Cluster dualities
  6. Relations to other works
  7. Preliminaries
  8. Semifields
  9. Mutations of matrices
  10. Generalized cluster algebras
  11. Generalized cluster algebras
  12. Relation to Chekhov--Shapiro's definition
  13. Principal coefficients
  14. Separation formula
  15. Cluster algebras with specialized coefficients
  16. ...and 30 more sections

Key Result

Theorem 1.1

There is a piecewise linear operation $T_k$ such that $T_k(\mathfrak D_\mathbf s)$ is equivalent to $\mathfrak D_{\mu_k(\mathbf s)}$ where $\mu_k(\mathbf s)$ denotes the mutation in direction $k$ of the seed $\mathbf s$.

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (155)

  • Theorem 1.1: \ref{['thm: mutation invariance']}
  • Theorem 1.2: \ref{['thm: wall crossing on cluster chamber']}
  • Theorem 1.3: \ref{['thm: cluster variable as theta function']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: fomin2002cluster
  • Definition 3.1
  • Definition 3.2
  • ...and 145 more