Y-frieze patterns

Abstract

Motivated by cluster ensembles, we introduce a new variant of frieze patterns associated to acyclic cluster algebras, which we call $\mathbf{Y}\text{-frieze patterns}$. Using the mutation rules for $\mathbf{Y}$-variables, we define a large class of $\mathbf{Y}$-frieze patterns called $\text{unitary }\mathbf{Y}\text{-frieze patterns}$, and show that the ensemble map induces a map from (unitary) frieze patterns to (unitary) $\mathbf{Y}$-frieze patterns. In rank 2, we show that $\mathbf{Y}$-frieze patterns are (associated to) friezes of generalised cluster algebras. In finite type (not necessarily rank 2), we show that $\mathbf{Y}$-frieze patterns share the same symmetries as frieze patterns, and prove that their number is finite.

arXiv commentSome notation changed, Section 4 on the case of rank 2 added. The finiteness conjecture of v1 is now Theorem 5.5