Frieze vectors and unitary friezes

Let $Q$ be a quiver without loops and $2$-cycles, let $\mathcal{A}(Q)$ be the corresponding cluster algebra and let $\mathbf{x}$ be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to $\mathbf{x}$. 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. Moreover, we show that every positive integral frieze of affine Dynkin type $\tilde{\mathbb{A}}_{p,q}$ 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.

Keywords

cluster algebra, frieze

2010 Mathematics Subject Classification

Primary 13F60. Secondary 16G20.

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The first-named author is supported by the NSF-CAREER grant DMS-1254567 and by the University of Connecticut.

The second-named author is supported by the NSF-CAREER grant DMS-1254567, the NSF grant DMS-1800860 and by the University of Connecticut.

Received 2 September 2018

Accepted 16 December 2019

Published 9 October 2020