[Paper Review] Arc representations

Salomón Domínguez|arXiv (Cornell University)|Sep 25, 2017

Manufacturing Process and Optimization2 citations

TL;DR

This paper extends Labardini-Fragoso's arc representations to tagged triangulations of surfaces with punctures, constructing explicit quiver with potential representations that satisfy Jacobian relations. It generalizes the arc representation framework to handle tagged arcs and triangulations, proving nilpotency and compatibility with quiver mutation via a conjecture on representation mutation.

ABSTRACT

This paper was inspired by four articles: surface cluster algebras studied by Fomin-Shapiro-Thurston \cite{fst}, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky \cite{dwz}, string modules associated to arcs on unpunctured surfaces by Assem-Br$\ddot{u}$stle-Charbonneau-Plamondon \cite{acbp} and Quivers with potentials associated to triangulated surfaces, part II: Arc representations by Labardini-Fragoso. \cite{lf2}. For a surface with marked points ($\Sigma,M$) Labardini-Fragoso associated a quiver with potential $(Q( au),S( au))$ then for an ideal triangulation of ($\Sigma,M$) and an ideal arc Labardini-Fragoso defined an arc representation of $(Q( au),S( au))$. This paper focuses on extent the definition of arc representation to a more general context by considering a tagged triangulation and a tagged arc. We associate in an explicit way a representation of the quiver with potential constructed Labardini-Fragoso and prove that the Jacobian relations are met.

Motivation & Objective

• Extend arc representations from ideal triangulations to tagged triangulations of surfaces with marked points and punctures.
• Address the failure of standard arc representations to satisfy Jacobian relations in the presence of punctures.
• Construct explicit representations of quivers with potentials associated to tagged triangulations that satisfy the Jacobian relations.
• Generalize the framework of Labardini-Fragoso to include tagged arcs and tagged triangulations.
• Conjecture that arc representations are compatible with quiver mutation under flip operations, enabling cluster character calculations.

Proposed method

• Define tagged triangulations and tagged arcs on surfaces with punctures, generalizing ideal triangulations.
• Construct the quiver with potential (Q(τ), S(τ)) from a tagged triangulation τ using the framework of Derksen-Weyman-Zelevinsky.
• Introduce detour curves and auxiliary curves to resolve Jacobian relation violations caused by punctures.
• Define the arc representation M(τ, i) using path algebra modules and detour matrices, ensuring compatibility with cyclic derivatives.
• Prove that the constructed representation satisfies all Jacobian relations via explicit computation of cyclic derivatives.
• Demonstrate nilpotency of the representation using admissibility and radical filtration in the path algebra quotient.

Experimental results

Research questions

• RQ1 How can arc representations be generalized from ideal to tagged triangulations in the presence of punctures?
• RQ2 Why do standard arc representations fail to satisfy Jacobian relations when punctures are present?
• RQ3 What role do detour curves and detour matrices play in restoring Jacobian consistency?
• RQ4 Is the arc representation M(τ, i) nilpotent, and how does this relate to the structure of the Jacobian algebra?
• RQ5 Can arc representations be compatible with quiver mutation under triangulation flips, as conjectured?

Key findings

• The paper constructs an explicit arc representation M(τ, i) for a tagged arc i in a tagged triangulation τ of a surface with punctures.
• The representation M(τ, i) satisfies all Jacobian relations of the quiver with potential (Q(τ), S(τ)), resolving a failure of standard representations in punctured settings.
• The representation M(τ, i) is proven to be nilpotent, as it lies in the radical filtration of the path algebra quotient.
• Explicit computation of cyclic derivatives ∂α(S(τ)) shows vanishing action on M(τ, i), confirming Jacobian consistency.
• The construction relies on detour curves and detour matrices to handle puncture-induced path ambiguities.
• A conjecture is proposed that arc representations commute with quiver mutation under flips, suggesting compatibility with cluster character theory.