Frieze varieties : A characterization of the finite-tame-wild trichotomy for acyclic quivers

TL;DR

This paper introduces frieze varieties associated with acyclic quivers and characterizes the finite-tame-wild classification of these quivers based on the dimension of their frieze varieties.

Contribution

It provides a new geometric characterization of the representation type of acyclic quivers using frieze varieties and their dimensions.

Findings

Finite quivers correspond to zero-dimensional frieze varieties.

Tame quivers have one-dimensional frieze varieties.

Wild quivers are associated with frieze varieties of dimension two or more.

Abstract

We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is $0,1$, or $\ge2$, respectively.