Séminaire Lotharingien de Combinatoire, 82B.74 (2019), 12 pp.

Emily Gunawan and Ralf Schiffler

Unitary friezes and frieze vectors

Abstract. We 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. For an arbitrary quiver Q, we introduce a new class of integer vectors which we call frieze vectors. These frieze vectors are defined as solutions of certain Diophantine equations given by the cluster variables in the cluster algebra. We establish a bijection between the positive unitary frieze vectors and the clusters in the cluster algebra.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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