Mutation of friezes

TL;DR

This paper explores how mutations in Conway-Coxeter friezes relate to cluster mutations in Dynkin type A categories, providing formulas for entry changes and submodule counts to better understand their combinatorial structure.

Contribution

It introduces a shape-based mutation formula for frieze entries and a combinatorial method to compute associated friezes from cluster-tilting objects.

Findings

A formula describing entry changes under mutation based on frieze shape

A division of the frieze into four mutation regions

A combinatorial formula for counting submodules of string modules

Abstract

We study mutations of Conway-Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type $A$. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type $A$ in the sense of Caldero and Chapoton.