$\tilde{A}$ and $\tilde{D}$ type cluster algebras: Triangulated surfaces and friezes

TL;DR

This paper characterizes all cluster variables in $ ilde{A}$ and $ ilde{D}$ type cluster algebras using triangulated surfaces, frieze patterns, and cluster categories, revealing their explicit structure and periodicity.

Contribution

It provides a complete description of cluster variables for these types via geometric and categorical methods, connecting frieze patterns with cluster algebra theory.

Findings

Cluster variables are expressed through frieze patterns and periodic quantities.

Cluster variables form friezes matching those in prior work.

The approach links geometric, algebraic, and categorical perspectives.

Abstract

By viewing $\tilde{A}$ and $\tilde{D}$ type cluster algebras as triangulated surfaces, we find all cluster variables in terms of either (i) the frieze pattern (or bipartite belt) or (ii) the periodic quantities previously found for the cluster map associated with these frieze patterns. We show that these cluster variables form friezes which are precisely the ones found in [1] by applying the cluster character to the associated cluster category.