Abstract

One approach to the representation theory of algebras is to study the module category of an algebra. This can be achieved, at least in part, by describing the indecomposable modules of an algebra and the irreducible morphisms between them. In some sense, these can be viewed as the building blocks for all modules and morphisms in the module category. The Auslander-Reiten quiver of an algebra is a means of presenting this information. Of particular interest is a class of algebras known as Brauer graph algebras. These are symmetric special biserial algebras that have a presentation in the form of a (decorated) ribbon graph called a Brauer graph. An interesting feature of Brauer graph algebras is that one can often read off aspects of the representation theory by performing a series of combinatorial games on the Brauer graph, which removes the need for potentially difficult and lengthy calculations. The purpose of this talk is show that one can read off information regarding the Auslander-Reiten theory of a Brauer graph algebra from its underlying Brauer graph. We begin by providing an algorithm for constructing the stable Auslander-Reiten component containing a given indecomposable module of a Brauer graph algebra using only information from its Brauer graph. We then 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.