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Mutation Cycles from Reddening Sequences

Tucker J. Ervin, Scott Neville

TL;DR

This work investigates how reddening sequences can be leveraged to generate mutation cycles in quiver mutation. By forming triangular extensions $Q = T \stackrel{A}{\rightarrow} H$ from two quivers with reddening sequences and concatenating these sequences, the authors obtain mutation cycles via $\mu_{\\mathbf M_T\\mathbf M_H}(Q)$, with exact equality or isomorphism indicating cycles. They establish conditions under which the cycles are simple using C-matrix-based distinguishability and analyze reddening sequences in forkless parts, including classifications in low rank for abundant acyclic quivers and keys. The results provide a versatile construction toolkit for producing many mutation cycles, offering new fully generic families, explicit long cycles, and non-extension examples, thereby enriching the understanding of cluster automorphisms and mutation equivalence in cluster algebras.

Abstract

Given two quivers, each with a reddening sequence, we show how to construct a plethora of mutation cycles. We give several examples, including a generalization of the construction of long mutation cycles in earlier work by the second author. We also give new results on the reddening sequences of certain mutation-acyclic quivers and forks, classifying them in some cases.

Mutation Cycles from Reddening Sequences

TL;DR

This work investigates how reddening sequences can be leveraged to generate mutation cycles in quiver mutation. By forming triangular extensions from two quivers with reddening sequences and concatenating these sequences, the authors obtain mutation cycles via , with exact equality or isomorphism indicating cycles. They establish conditions under which the cycles are simple using C-matrix-based distinguishability and analyze reddening sequences in forkless parts, including classifications in low rank for abundant acyclic quivers and keys. The results provide a versatile construction toolkit for producing many mutation cycles, offering new fully generic families, explicit long cycles, and non-extension examples, thereby enriching the understanding of cluster automorphisms and mutation equivalence in cluster algebras.

Abstract

Given two quivers, each with a reddening sequence, we show how to construct a plethora of mutation cycles. We give several examples, including a generalization of the construction of long mutation cycles in earlier work by the second author. We also give new results on the reddening sequences of certain mutation-acyclic quivers and forks, classifying them in some cases.

Paper Structure

This paper contains 8 sections, 35 theorems, 34 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and prior work
  3. Mutation cycles from reddening sequences
  4. Reddening Sequences and Forks
  5. Reddening Sequences in Low Rank
  6. Examples
  7. New mutation cycles
  8. Other cycles

Key Result

Proposition 2.3

Let $Q$ be a quiver with a vertex $i$.

Figures (26)

  • Figure 1: A triangular extension of quivers of finite types $A_2$ (with vertices $\{5,6\}$) and $D_4$ (with vertices $\{1,2,3,4\}$). Mutating at vertices $5,6, 1, 2, 1, 3, 2, 4, 2, 1$ gives a mutation cycle.
  • Figure 2: Two isomorphic but not equal quivers. An isomorphism is given by exchanging 1 and 2.
  • Figure 3: The principle framing of an oriented $3$-cycle.
  • Figure 4: A reddening sequence of a $3$-vertex quiver.
  • Figure 5: Another reddening sequence of a $3$-vertex quiver.
  • ...and 21 more figures

Theorems & Definitions (104)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: Sign-Coherence MR2629987
  • Definition 2.9
  • Definition 2.10
  • ...and 94 more