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Invariant Functions On Cluster Ensembles

Dani Kaufman

Abstract

We define the notion of an invariant function on a cluster ensemble with respect to an action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We show that these invariants have geometric and number theoretic interpretations, and classify them for ensembles associated to affine Dynkin diagrams.

Invariant Functions On Cluster Ensembles

Abstract

We define the notion of an invariant function on a cluster ensemble with respect to an action of the cluster modular group on its associated function fields. We realize many examples of previously studied functions as elements of this type of invariant ring and give many new examples. We show that these invariants have geometric and number theoretic interpretations, and classify them for ensembles associated to affine Dynkin diagrams.

Paper Structure

This paper contains 28 sections, 9 theorems, 58 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Structure of this paper
  3. Background
  4. Quivers
  5. Positive spaces
  6. The $\mathcal{A}$ space
  7. The $\mathcal{X}$ space
  8. The type of a cluster ensemble
  9. The cluster modular group
  10. Action of the cluster modular group on functions
  11. Invariants of the cluster modular group
  12. Examples
  13. Basic properties
  14. Trivial Invariants
  15. Invariants of Subensembles
  16. ...and 13 more sections

Key Result

Theorem 4.1

$\mathbb{R}(\mathcal{A}_Q)^{<\gamma_Q>}$ is exactly $\mathbb{R}(\mathcal{A}_R)^{<\gamma_R>}$. Furthermore, evaluating the variables associated to the nodes in $N(Q)-N(R^\mu)$ at 1 gives a map of sets $\mathbb{R}(\mathcal{A}_Q)^{<\gamma_Q>}\cup\{\infty\} \rightarrow \mathbb{R}(\mathcal{A}_{R^\mu})^{<

Figures (16)

  • Figure 1: Two quivers of type $\widetilde{G}_2$
  • Figure 2: Choices of quiver isomorphism classes in the mutation class of a quiver of type $A_3$.
  • Figure 3: The cluster modular groupoid of the $A_3$ cluster ensemble. A selection of non-identity maps are shown.
  • Figure 4: Quiver of type $\widetilde{A}_1$.
  • Figure 5: Quivers associated with doubly extended Dynkin diagrams with 3 nodes.
  • ...and 11 more figures

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.1
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 51 more