Gentle algebras arising from surfaces with orbifold points of order 3, Part I: scattering diagrams

TL;DR

This paper explores the connection between gentle algebras derived from surfaces with orbifold points of order three and their associated scattering diagrams, establishing a bijection between certain algebraic objects and cluster monomials.

Contribution

It introduces a new class of gentle algebras from surfaces with orbifold points and proves a bijection between $ au$-rigid pairs and cluster monomials using scattering diagrams.

Findings

Finite-dimensional gentle algebras from orbifold surfaces are constructed.

Stability scattering diagrams are used to analyze these algebras.

A bijection between $ au$-rigid pairs and cluster monomials is established.

Abstract

To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero--Chapoton map defines a bijection between reachable $\tau$-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.