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Mutations of quivers with 2-cycles

Fang Li, Siyang Liu, Lang Mou, Jie Pan

TL;DR

The paper develops a mutation theory for quivers with oriented $2$-cycles by introducing a homotopy $H$, a normal subgroupoid of the quiver's fundamental groupoid, enabling involutive mutations and invariant quotient groupoids. It unifies Fomin–Zelevinsky mutations, orbit mutations from coverings, and surface-triangulation constructions by establishing a flip–mutation correspondence for quivers with $2$-cycles and constructing associated 2-cycle cluster algebras. It shows how to realize quivers with homotopies from colored puncture triangulations, including a topological realization via a $2$-complex, and extends the surface model of Fomin–Shapiro–Thurston to this setting. The Laurent phenomenon is proven in cases arising from globally weakly admissible coverings with infinite orbit mutations, and several future directions are discussed, including potential links to DWZ mutations and categorifications.

Abstract

We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles.

Mutations of quivers with 2-cycles

TL;DR

The paper develops a mutation theory for quivers with oriented -cycles by introducing a homotopy , a normal subgroupoid of the quiver's fundamental groupoid, enabling involutive mutations and invariant quotient groupoids. It unifies Fomin–Zelevinsky mutations, orbit mutations from coverings, and surface-triangulation constructions by establishing a flip–mutation correspondence for quivers with -cycles and constructing associated 2-cycle cluster algebras. It shows how to realize quivers with homotopies from colored puncture triangulations, including a topological realization via a -complex, and extends the surface model of Fomin–Shapiro–Thurston to this setting. The Laurent phenomenon is proven in cases arising from globally weakly admissible coverings with infinite orbit mutations, and several future directions are discussed, including potential links to DWZ mutations and categorifications.

Abstract

We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles.

Paper Structure

This paper contains 24 sections, 22 theorems, 62 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Generalizing orbit mutations
  4. Quivers with homotopies arising from surface triangulations
  5. Cluster algebras associated to quivers with $2$-cycles
  6. Quiver mutation and covering of quivers
  7. The Fomin--Zelevinsky quiver mutation
  8. Coverings of quivers
  9. Orbit mutations
  10. Quivers with homotopies
  11. Quivers with homotopies
  12. Mutations of quivers with homotopies
  13. Mutation is an involution
  14. Patterns of quivers with homotopies
  15. Homotopies and orbit mutations
  16. ...and 9 more sections

Key Result

Proposition 2.8

There is a bijection between regular coverings of $Q$ (up to isomorphisms relative to $Q$) and normal subgroups of $\pi_1(Q, v)$ for any $v\in Q_0$. If $p \colon \widetilde{Q} \rightarrow Q$ is a regular covering, the fundamental group $\pi_1(Q, v)$ for any $v\in Q_0$ acts on $\widetilde{Q}$ by deck

Figures (15)

  • Figure 1: Mutations with different homotopies result in different quivers.
  • Figure 2: A globally weakly admissible covering.
  • Figure 3: A globally weakly admissible covering.
  • Figure 4: From left to right: three puzzle pieces $P_1, P_2, P_3$ and their associated quivers.
  • Figure 5: The new puzzle piece $P_4$ and its associated quiver.
  • ...and 10 more figures

Theorems & Definitions (89)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Proposition 2.8: Galois correspondence
  • Example 2.9
  • Definition 2.10
  • ...and 79 more