Friezes and a construction of the euclidean cluster variables

TL;DR

This paper introduces an algorithm leveraging friezes to compute cluster variables and characters in Euclidean quiver cluster categories, linking algebraic and geometric invariants.

Contribution

It provides a novel algorithm for calculating cluster characters and variables in Euclidean quivers using friezes, connecting algebraic and geometric data.

Findings

Algorithm computes all cluster variables for Euclidean quivers.

Allows calculation of Euler characteristics of quiver Grassmannians.

Bridges cluster algebra computations with geometric invariants.

Abstract

Let $Q$ be an euclidean quiver. Using friezes in the sense of Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical) cluster character associated to any object in the cluster category of $Q$. In particular, this algorithm allows to compute all the cluster variables in the cluster algebra associated to $Q$. It also allows to compute the sum of the Euler characteristics of the quiver grassmannians of any module $M$ over the path algebra of $Q$.