arXiv subjects
Publications and source records attributed to Antoine de Saint Germain.
Inspired by a recent work of Y. Mizuno, we show that the DT transformations on a cluster ensemble of finite type admit unique totally positive fixed points, and that the exponents of the linearizations of the DT transformations at the fixed points are precisely the degrees of the Weyl group of the corresponding finite root system.
In this note, we show that the sequence of maximum values in frieze patterns of type $A_n$ is the sequence of Fibonacci numbers, and that of frieze patterns of type $C_n$ is the sequence of odd Fibonacci numbers.
We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.
We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.
We give an elementary account of the notion of Y-frieze patterns, explain some of their properties, and reveal their connection with Coxeter's frieze patterns.
Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call ${\bf Y}\textit{-frieze patterns}$. Using the mutation rules for ${\bf Y}$-variables, we define a large class of ${\bf Y}$-frieze patterns called $\textit{unitary }{\bf Y}\textit{-frieze patterns}$, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) ${\bf Y}$-frieze patterns. In rank 2, we show that ${\bf Y}$-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that ${\bf Y}$-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite.
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.