Derived equivalences between symmetric special biserial algebras

TL;DR

This paper introduces a combinatorial approach to tilting mutation in symmetric special biserial algebras, generalizing Rickard's star theorem and applying to Brauer graph algebras.

Contribution

It provides a new combinatorial description of tilting mutation for symmetric special biserial algebras and extends existing theorems to Brauer graph algebras.

Findings

Explicit combinatorial description of tilting mutation

Generalization of Rickard's star theorem

Application to Brauer graph algebras

Abstract

The notion of mutation plays crucial roles in representation theory of algebras. Two kinds of mutation are well-known: tilting/silting mutation and quiver-mutation. In this paper, we focus on tilting mutation for symmetric algebras. Introducing mutation of SB quivers, we explicitly give a combinatorial description of tilting mutation of symmetric special biserial algebras. As an application, we generalize Rickard's star theorem. We also introduce flip of Brauer graphs and apply our results to Brauer graph algebras.