Tubular Cluster Algebras II: Exponential growth

TL;DR

This paper proves that all tubular (simply laced) mutation finite cluster algebras exhibit exponential growth, using automorphism group analysis and explicit mutation sequences.

Contribution

It establishes the exponential growth of tubular cluster algebras through two novel methods, enhancing understanding of their algebraic complexity.

Findings

Tubular cluster algebras are of exponential growth

Automorphism groups help determine growth rates

Explicit mutation sequences demonstrate growth behavior

Abstract

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding cluster category and second by giving explicit sequences of mutations.