In the paper under review, the authors introduce a new class of algebraic varieties called the frieze varieties. The frieze variety is defined in an elementary recursive way by constructing a set of points in the 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. Note that each frieze variety is determined by an acyclic quiver.
It is well known that the acyclic quiver is representation finite if and only if its underlying graph is a Dynkin diagram of type \(\mathbb{A}\), \(\mathbb{D}\) or \(\mathbb{E}\), and it is tame if and only if the underlying graph is an affine Dynkin diagram of type \(\widetilde{\mathbb{A}}\), \(\widetilde{\mathbb{D}}\) or \(\widetilde{\mathbb{E}}\). All other acyclic quivers are wild.
In this paper, the authors give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of their frieze varieties.
More precisely, they prove that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its friezevariety is \(0\), \(1\), or \(\geq 2\), respectively.
Finally, let us mention that there are several characterizations of the finite-tame-wild trichotomy in the literature, however, it seems that the characterization given by the authors is the first one in terms of numerical invariants that are integers.

Reviewer: Piotr Malicki (Toruń)

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