# Cluster tilting modules for mesh algebras

## TL;DR

This paper investigates cluster tilting modules in mesh algebras of Dynkin type, establishing their existence, properties, and mutation behavior, including a novel example outside the stably 2-Calabi-Yau setting.

## Contribution

It provides a new proof of the existence of cluster tilting modules and characterizes their maximal rigid modules and mutation in mesh algebras.

## Key findings

- Cluster tilting modules are precisely the maximal rigid modules in most cases.
- These modules are equivariant under a specific automorphism.
- An explicit example of mutation in a non-stably 2-Calabi-Yau abelian category is given.

## Abstract

We study cluster tilting modules in mesh algebras of Dynkin type, providing a new proof for their existence. In all but one case, we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics.