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Geometry of $C$-vectors and $C$-Matrices for Mutation-Infinite Quivers

Tucker J. Ervin, Blake Jackson, Kyungyong Lee, Son Dang Nguyen

Abstract

The set of forks is a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let $Q$ be a fork with $n$ vertices, and $\boldsymbol{w}$ be a fork-preserving mutation sequence. We show that every $c$-vector of $Q$ obtained from $\boldsymbol{w}$ is a solution to a quadratic equation of the form $$\sum_{i=1}^n x_i^2 + \sum_{1\leq i<j\leq n} \pm q_{ij} x_i x_j =1,$$ where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. The same proof techniques implies that when $Q$ is a rank 3 mutation-cyclic quiver, every $c$-vector of $Q$ is a solution to a quadratic equation of the same form.

Geometry of $C$-vectors and $C$-Matrices for Mutation-Infinite Quivers

Abstract

The set of forks is a class of quivers introduced by M. Warkentin, where every connected mutation-infinite quiver is mutation equivalent to infinitely many forks. Let be a fork with vertices, and be a fork-preserving mutation sequence. We show that every -vector of obtained from is a solution to a quadratic equation of the form where is the number of arrows between the vertices and in . The same proof techniques implies that when is a rank 3 mutation-cyclic quiver, every -vector of is a solution to a quadratic equation of the same form.

Paper Structure

This paper contains 11 sections, 34 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C$-matrices
  4. Reflections and $L$-matrices
  5. Geometry of reflections
  6. Forks and Admissible GIM's
  7. Generalized intersection matrices for mutated quivers
  8. $C$-Vectors And $L$-Vectors
  9. Proof of Theorem \ref{['thm-coxeter-element']}
  10. Examples
  11. More examples

Key Result

Theorem 1.1

Let $n$ be any positive integer. Let $P$ be a mutation-infinite connected quiver with $n$ vertices. Then there exist an infinite number of pairs of a quiver $Q\in \text{Mut}(P)$ and $k\in\{1,...,n\}$ such that every $c$-vector of $Q$ obtained from any mutation sequence not starting with $k$ is a sol where $q_{ij}$ is the number of arrows between the vertices $i$ and $j$ in $Q$. There does not seem

Figures (1)

  • Figure 1: This picture illustrates a portion of the universal cover $\Sigma_\sigma$, and the three arcs $T_\alpha, T_\beta,$ and $T_\gamma$.

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • Corollary 1.11
  • ...and 56 more