Auslander-Reiten Components of Symmetric Special Biserial Algebras

TL;DR

This paper introduces a combinatorial method to construct and analyze the stable Auslander-Reiten components of symmetric special biserial algebras using Brauer graphs, revealing their structural relationship with Green walks.

Contribution

It provides a new algorithmic approach to determine Auslander-Reiten components from Brauer graph data, linking graph structure to module categories in symmetric special biserial algebras.

Findings

The structure of Auslander-Reiten quivers relates to Green walks on Brauer graphs.

The shape of stable Auslander-Reiten components for domestic Brauer graph algebras depends on the underlying graph.

The component containing a simple or indecomposable projective module is determined by the associated edge in the Brauer graph.

Abstract

We provide a combinatorial algorithm for constructing the stable Auslander-Reiten component containing a given indecomposable module of a symmetric special biserial algebra using only information from its underlying Brauer graph. We also show that the structure of the Auslander-Reiten quiver is closely related to the distinct Green walks around the Brauer graph and detail the relationship between the precise shape of the stable Auslander-Reiten components for domestic Brauer graph algebras and their underlying graph. Furthermore, we show that the specific component containing a given simple or indecomposable projective module for any Brauer graph algebra is determined by the edge in the Brauer graph associated to the module.