Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Mysterious points in keys but not trees

Scott Neville, José Simental

Abstract

The deep locus of a cluster variety is defined to be the set of its points that do not belong to any cluster torus. We show that, if the cluster variety has a seed whose mutable part is a tree without multiple edges, then the deep locus can be characterized as the set of points whose stabilizer under a certain group action is nontrivial. Deep points without a stabilizer are called mysterious. We establish that many other classes of acyclic quivers (including keys) often have mysterious points. This refutes Conjecture 1.1 of arXiv:2402.16970, but establishes it in many important cases.

Mysterious points in keys but not trees

Abstract

The deep locus of a cluster variety is defined to be the set of its points that do not belong to any cluster torus. We show that, if the cluster variety has a seed whose mutable part is a tree without multiple edges, then the deep locus can be characterized as the set of points whose stabilizer under a certain group action is nontrivial. Deep points without a stabilizer are called mysterious. We establish that many other classes of acyclic quivers (including keys) often have mysterious points. This refutes Conjecture 1.1 of arXiv:2402.16970, but establishes it in many important cases.
Paper Structure (16 sections, 19 theorems, 31 equations, 1 figure)

This paper contains 16 sections, 19 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cluster varieties
  3. Cluster algebras
  4. Cluster varieties and the deep locus
  5. The cluster dilation group
  6. Acyclic case
  7. Tree cluster algebras
  8. Statement of the theorem
  9. Reductions
  10. Proof of Theorem \ref{['thm:main-notintro']}
  11. Deep points from rank 2 subalgebras
  12. Mysterious points
  13. Keys
  14. Forks
  15. Locally acyclic
  16. ...and 1 more sections

Key Result

Theorem 1.1

Assume $\mathcal{A}$ has a seed $\Sigma = (\widetilde{B}(Q), \widetilde{\mathbf{x}})$ such that the mutable part of $Q$ is a tree. Then, $\mathcal{A}$ has no mysterious points.

Figures (1)

  • Figure 1: An example of the sort of quivers we consider to prove Theorem \ref{['thm:main-notintro']}. Note that every vertex is either a sink or a source. Frozen vertices are marked as boxes.

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 46 more