Friezes with coefficients

For a cluster algebra $\mathcal{A}$ over $\mathbb{Q}$ of geometric type, we define a $\textit{frieze}$ of $\mathcal{A}$ 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. For cluster algebras with acyclic seeds, we introduce $\textit{frieze patterns with coefficients}$ and we give a sufficient condition for 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.Comment: 47 page

Tropical patterns from the Fock-Goncharov pairing and Ringel's conjectures

We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. As a special case, we generalize and prove two conjectures of C. M. Ringel on cluster-additive functions for arbitrary Cartan matrices of finite type