Friezes of cluster algebras of geometric type

TL;DR

This paper studies friezes of cluster algebras of geometric type, exploring their properties, criteria, and connections to Lie theory, and extends classical frieze pattern symmetries to finite types with various coefficients.

Contribution

It introduces the concept of friezes for cluster algebras, generalizes frieze patterns to acyclic seeds, and links frieze points to geometric and Lie-theoretic structures.

Findings

Frieze testing criteria established.

Frieze points characterized geometrically and Lie-theoretically.

Symmetry of finite type frieze patterns determined.

Abstract

For a cluster algebra $\mathcal{A}$ over $\mathbb{Q}$ of geometric type, a $\textit{frieze}$ of $\mathcal{A}$ is defined to be a $\mathbb{Q}$-algebra homomorphism from $\mathcal{A}$ to $\mathbb{Q}$ that takes positive integer values on all cluster variables and all frozen variables. We present some basic facts on friezes, including frieze testing criteria, the notion of $\textit{frieze points}$ when $\mathcal{A}$ is finitely generated, and pullbacks of friezes under certain $\mathbb{Q}$-algebra homomorphisms. When the cluster algebra $\mathcal{A}$ is acyclic, we define $\textit{frieze patterns associated to acyclic seeds of }\mathcal{A}$, generalizing the $\textit{ frieze patterns with coefficients of type } A$ studied by J. Propp and by M. Cuntz, T. Holm, and P. Jorgensen, and we give a sufficient condition for such frieze patterns to be equivalent to friezes. For the special cases when $\mathcal{A}$ has an acyclic seed with either trivial coefficients, principal coefficients, or what we call the $\textit{BFZ coefficients}$ (named after A. Berenstein, S. Fomin, and A. Zelevinsky), we identify frieze points of $\mathcal{A}$ both geometrically as certain positive integral points in explicitly described affine varieties and Lie theoretically (in the finite case) in terms of reduced double Bruhat cells and generalized minors on the associated semi-simple Lie groups. Furthermore, extending the gliding symmetry of the classical Coxeter frieze patterns of type $A$, we determine the symmetry of frieze patterns of any finite type with arbitrary coefficients.