Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quivers with potentials associated to triangulated surfaces, Part II: Arc representations

Daniel Labardini-Fragoso

TL;DR

The paper develops arc representations M(τ,i) of quivers with potentials (Q(τ), S(τ)) attached to arcs on bordered surfaces and proves that flips of ideal triangulations correspond to mutations of these QP-representations, thereby providing a representation-theoretic realization of surface cluster dynamics. Arc representations are constructed via detours and detour matrices that twist a base segment representation, and are shown to satisfy Jacobian relations and to be mutation-equivalent to negative simples, enabling explicit computation of F-polynomials and g-vectors in the positive stratum. The results connect the geometric model of surface clusters with the DWZ mutation framework, extend the ABCP string-module viewpoint to punctured settings, and yield practical formulas for cluster invariants while highlighting several open problems in non-degeneracy, tagged triangulations, and combinatorial descriptions of Jacobian data.

Abstract

This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation.

Quivers with potentials associated to triangulated surfaces, Part II: Arc representations

TL;DR

The paper develops arc representations M(τ,i) of quivers with potentials (Q(τ), S(τ)) attached to arcs on bordered surfaces and proves that flips of ideal triangulations correspond to mutations of these QP-representations, thereby providing a representation-theoretic realization of surface cluster dynamics. Arc representations are constructed via detours and detour matrices that twist a base segment representation, and are shown to satisfy Jacobian relations and to be mutation-equivalent to negative simples, enabling explicit computation of F-polynomials and g-vectors in the positive stratum. The results connect the geometric model of surface clusters with the DWZ mutation framework, extend the ABCP string-module viewpoint to punctured settings, and yield practical formulas for cluster invariants while highlighting several open problems in non-degeneracy, tagged triangulations, and combinatorial descriptions of Jacobian data.

Abstract

This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation.

Paper Structure

This paper contains 18 sections, 30 theorems, 221 equations, 36 figures.

Table of Contents

  1. Introduction
  2. Background on QP-representations and their mutations
  3. Quivers with potentials and their mutations
  4. Decorated representations and their mutations
  5. Restriction
  6. The QP of an ideal triangulation
  7. Definition of arc representations
  8. First case: $i$ does not cut out a once-punctured monogon
  9. Second case: $i$ cuts out a once-punctured monogon
  10. Checking Jacobian relations
  11. Local decompositions
  12. $M(\tau,i)$ satisfies the Jacobian ideal
  13. Flip $\leftrightarrow$ mutation compatibility
  14. Effect of flips on detour matrices
  15. Main result: Statement and proof
  16. ...and 3 more sections

Key Result

Theorem 2.2

For every QP $(Q,S)$ there exist a trivial QP $(Q_{\operatorname{triv}},S_{\operatorname{triv}})$ and a reduced QP $(Q_{\operatorname{red}},S_{\operatorname{red}})$ such that $(Q,S)$ is right-equivalent to the direct sum $(Q_{\operatorname{triv}},S_{\operatorname{triv}})\oplus(Q_{\operatorname{red}}

Figures (36)

  • Figure 1: Paths are composed as functions
  • Figure 2: Self-folded triangle
  • Figure 3:
  • Figure 4: Drawing $n+1$-detours after drawing the $n$-detours
  • Figure 5:
  • ...and 31 more figures

Theorems & Definitions (83)

  • Definition 2.1
  • Theorem 2.2: Splitting Theorem, DWZ, Theorem 4.6
  • Definition 2.3: DWZ, equations (5.3) and (5.8) and Definition 5.5
  • Theorem 2.4: DWZ, Theorem 5.2 and Corollary 5.4
  • Theorem 2.5: DWZ, Theorem 5.7
  • Definition 2.6: DWZ, Definition 7.2
  • Theorem 2.7: DWZ, Proposition 7.3 and Corollary 7.4
  • Theorem 2.8: DWZ, Corollary 6.11, Proposition 8.1 and Corollary 8.2
  • Proposition 2.9: DWZ, Corollary 6.6
  • Definition 2.10: DWZ, Definition 10.1
  • ...and 73 more