Growth of infinite frieze patterns of affine type

TL;DR

This paper investigates the growth behavior of infinite frieze patterns associated with affine type cluster algebras, providing explicit formulas and demonstrating uniform growth coefficients across different tubes.

Contribution

It introduces a method to compute growth coefficients of infinite frieze patterns in affine types and proves their uniformity across stable tubes.

Findings

All infinite frieze patterns in affine type have the same growth coefficients.

An explicit formula for the k-th growth coefficient is derived.

The formula relates growth coefficients to data from homogeneous tubes and cluster algebra elements.

Abstract

We analyse the growth coefficients of infinite frieze patterns arising from cluster algebras using cluster modular groups and cluster categories. For a fixed cluster category of affine type, we prove that the collection of infinite frieze patterns given by both the homogeneous and non-homogeneous stable tubes all have the same growth coefficients. We also derive and verify an explicit formula for the $k$-th growth coefficient, expressed directly in terms of data from homogeneous tubes, or, alternatively, from appropriate elements of the corresponding cluster algebra.