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# Journal of Combinatorics

## Volume 11 (2020)

### Number 4

### Frieze vectors and unitary friezes

Pages: 681 – 703

DOI: https://dx.doi.org/10.4310/JOC.2020.v11.n4.a6

#### Authors

#### Abstract

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.

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