Arc representations

TL;DR

This paper extends the concept of arc representations in surface cluster algebras to include tagged triangulations and arcs, providing explicit constructions and verifying Jacobian relations.

Contribution

It generalizes arc representations to tagged triangulations and arcs, expanding the applicability of quiver with potential models in surface cluster algebras.

Findings

Explicit construction of arc representations for tagged arcs.

Proof that Jacobian relations are satisfied in the generalized setting.

Extension of previous models to more complex surface configurations.

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 $(Q(\tau),S(\tau))$ then for an ideal triangulation of ($\Sigma,M$) and an ideal arc Labardini-Fragoso defined an arc representation of $(Q(\tau),S(\tau))$. 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.