# Frieze varieties are invariant under Coxeter mutation

## TL;DR

This paper introduces a generalized frieze variety linked to acyclic quivers and demonstrates its invariance under Coxeter mutation, revealing deep symmetries in cluster algebra structures.

## Contribution

It defines a generalized frieze variety and proves its invariance under Coxeter mutation, expanding understanding of cluster algebra symmetries.

## Key findings

- Generalized frieze variety determined by any generic element.
- Coxeter mutation cyclically permutes components of the variety.
- Introduction of invariant Laurent polynomial technique.

## Abstract

We define a generalized version of the frieze variety introduced by Lee, Li, Mills, Seceleanu and the second author. The generalized frieze variety is an algebraic variety determined by an acyclic quiver and a generic specialization of cluster variables in the cluster algebra for this quiver. The original frieze variety is obtained when this specialization is (1, . . . , 1). The main result is that a generalized frieze variety is determined by any generic element of any component of that variety. We also show that the "Coxeter mutation" cyclically permutes these components. In particular, this shows that the frieze variety is invariant under the Coxeter mutation at a generic point. The paper contains many examples which are generated using a new technique which we call an invariant Laurent polynomial. We show that a symmetry of a mutation of a quiver gives such an invariant rational function.