Frieze Vectors and Unitary Friezes

TL;DR

This paper introduces frieze vectors in cluster algebras, showing they uniquely determine clusters and proving all positive integral friezes of affine Dynkin type A are unitary, thus completing the unitarity classification.

Contribution

It defines frieze vectors as solutions to Diophantine equations in cluster algebras and proves their role in determining clusters and unitarity of positive integral friezes.

Findings

Every cluster yields a frieze vector.

Frieze vectors determine the corresponding cluster.

All positive integral friezes of affine Dynkin type A are unitary.

Abstract

Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We show that every cluster gives rise to a frieze vector and that the frieze vector determines the cluster. We also study friezes of type Q as homomorphisms from the cluster algebra to an arbitrary integral domain. In particular, we show that every positive integral frieze of affine Dynkin type A is unitary, which means it is obtained by specializing each cluster variable in one cluster to the constant 1. This completes the answer to the question of unitarity for all positive integral friezes of Dynkin and affine Dynkin types.