# On a family of Caldero-Chapoton algebras that have the Laurent phenomenon

## TL;DR

This paper constructs a family of generalized cluster algebras as Caldero-Chapoton algebras associated with quivers derived from polygons with an orbifold point, establishing their algebraic and combinatorial properties.

## Contribution

It introduces a new geometric realization of generalized cluster algebras via Caldero-Chapoton algebras linked to polygons with orbifold points, connecting algebraic and combinatorial structures.

## Key findings

- Realization of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations.
- Establishment of a bijection between cluster variables and isomorphism classes of certain representations.
- Verification that exchange relations correspond to products of Caldero-Chapoton functions.

## Abstract

We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc $j$ on the polygon with orbifold point a representation $M(j)$ of the referred quiver with relations, and by proving that for every triangulation $\tau$ and every arc $j\in\tau$, the product of the Caldero-Chapoton functions of $M(j)$ and $M(j')$, where $j'$ is the arc that replaces $j$ when we flip $j$ in $\tau$, equals the corresponding exchange polynomial of Chekhov-Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of $E$-rigid indecomposable decorated representations of $\Lambda$.