On the growth of friezes via theta functions

TL;DR

This paper proves that all infinite friezes from tubes of a specific class of cluster algebras share the same growth coefficients, using identities of theta functions, extending prior affine type results.

Contribution

It establishes a unifying property of growth coefficients for infinite friezes in affine cluster algebras using theta function identities.

Findings

All infinite friezes from tubes of a given affine cluster algebra have identical growth coefficients.

The proof employs identities satisfied by theta functions.

Generalizes previous affine type results to a broader class.

Abstract

We prove that the infinite friezes arising from the tubes of a given cluster algebra of acyclic affine type all have the same growth coefficients. Our proof uses identities satisfied by theta functions. This generalizes previous results in affine types~$ADE$ by several groups of authors.