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Boundedness criteria for real quivers of rank 3

Roger Casals, Kenton Ke

TL;DR

This paper investigates boundedness of mutation classes for quivers with real weights in rank 3. It defines the relevant norms and the Markov constant $C(Q)$ and proves a sharp criterion: for $Q=(p,q,r)$ with $p\ge q\ge r\ge 0$, the mutation class $[Q]$ is bounded if and only if $p\le 2$ and $C(Q)\le 4$, with two distinct proofs—an explicit analytic bound-based argument and a geometric proof via Felikson–Tumarkin models. The results also yield a practical bound $\|[Q]\|\le\sqrt{C(Q)}$ in the bounded case and demonstrate unboundedness in mutation-infinite classes when $C(Q)>4$ or certain weight configurations exist. The paper thus provides a complete, verifiable criterion for boundedness in rank 3, connects mutation dynamics to hyperbolic and line-geometry realizations, and offers an appendix with SageMath code to visualize mutation sequences and figures. These findings contribute to understanding the dynamical landscape of real-weight quivers and open avenues for extending the characterization to higher ranks and exploring mutation-dynamics properties.

Abstract

We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.

Boundedness criteria for real quivers of rank 3

TL;DR

This paper investigates boundedness of mutation classes for quivers with real weights in rank 3. It defines the relevant norms and the Markov constant and proves a sharp criterion: for with , the mutation class is bounded if and only if and , with two distinct proofs—an explicit analytic bound-based argument and a geometric proof via Felikson–Tumarkin models. The results also yield a practical bound in the bounded case and demonstrate unboundedness in mutation-infinite classes when or certain weight configurations exist. The paper thus provides a complete, verifiable criterion for boundedness in rank 3, connects mutation dynamics to hyperbolic and line-geometry realizations, and offers an appendix with SageMath code to visualize mutation sequences and figures. These findings contribute to understanding the dynamical landscape of real-weight quivers and open avenues for extending the characterization to higher ranks and exploring mutation-dynamics properties.

Abstract

We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.

Paper Structure

This paper contains 13 sections, 7 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Scientific context
  3. Main result
  4. First proof of the main results
  5. A preliminary characterization
  6. Two quick lemmas
  7. Proof of \ref{['thm:main']}
  8. Proof of \ref{['thm:detbdd']}
  9. Alternative Proof of Main Result
  10. Final remarks and questions
  11. Appendix: SageMath Code and details for figures
  12. SageMath code to generate and plot random mutation sequences of a quiver
  13. Mutation sequences for Figures \ref{['fig:Examples1']} and \ref{['fig:Examples2']}

Key Result

Theorem 1.1

Let $Q$ be a quiver with real weights $(p,q,r)$ with $p\geq q\geq r\geq 0$. Then

Figures (3)

  • Figure 1: Depiction of the $(p,q,r)\in\mathbb R^3$ coordinates for three quivers $Q$ and a random sequence of mutations applied to them. In these cases, all three mutation classes $[Q]$ are bounded. The mutation sequences have been generated at random and are provided in the appendix. In the main text the variables $p,q,r\in\mathbb R_{\geq0}$ are always non-negative but, to ease implementation, the code and the figures it produces have $(p,q,r)\in\mathbb R^3$. In this case, a negative sign can readily be transformed into a positive sign by reversing the corresponding arrow of the quiver: the arrow goes from vertex 1 to vertex 2, if $p$ positive, or from vertex 2 to vertex 1, if $p$ is negative.
  • Figure 2: Depiction of the $(p,q,r)\in\mathbb R^3$ coordinates for quivers obtain by applying three different sequences to $Q=(-0.6,-0.43,0.567)$. The mutation class $[Q]$ is bounded. The mutation sequences have been generated at random, cf. \ref{['sec:appendix']}.
  • Figure 3: Depiction of the geometric realization of a mutation-cyclic rank $3$ quiver, where we are applying alternating mutations of $a_1,a_2$, realized by $\pi$-rotations, in order to translate them far away from $a_3$. In this figure, the points $a_1',a_2',a_1",\dots$ are obtained by $\pi$-reflecting along $a_1$ and $a_2$ and iterating.

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Claim 1
  • proof : Proof of \ref{['claim1']}
  • Lemma 2.3
  • proof
  • ...and 4 more