Three kinds of mutation

TL;DR

This paper compares mutation operations across three related concepts—exceptional sequences, silting objects, and m-cluster tilting objects—in the context of finite dimensional hereditary algebras, revealing their similarities and differences.

Contribution

It introduces and analyzes mutation operations for silting objects and compares them with existing mutations on exceptional sequences and m-cluster tilting objects.

Findings

Mutation operations are defined for silting objects.

Mutations on exceptional sequences and m-cluster tilting objects are compared.

The relationships between these mutation notions are clarified.

Abstract

For a finite dimensional hereditary algebra, we consider: exceptional sequences in the category of finite dimensional modules, silting objects in the bounded derived category, and m-cluster tilting objects in the m-cluster category. There are mutation operations on both the set of m-cluster tilting objects and the set of exceptional sequences. It is also possible to define a mutation operation for silting objects. We compare these three different notions of mutation.