On the Module Category of Symmetric Special Multiserial Algebras

The module category of an algebra is a major source of study for representation theorists. The indecomposable modules over an algebra and the morphisms between them are of tremendous importance, since these essentially determine the finitely generated module category over the algebra. The Auslander-Reiten quiver is a means of presenting this information. In this thesis, we focus on the class of symmetric special multiserial algebras. These are a broad class of algebras that include the well-studied subclass of symmetric special biserial algebras. A useful property of these algebras is that they have a decorated hypergraph (with orientation) associated to them, called a Brauer configuration. As well as offering a pictorial presentation of the algebra, many aspects of the representation theory are encoded in the combinatorial data of the hypergraph. In the first half of this thesis, we show that the Auslander-Reiten quiver of a symmetric special biserial algebra is completely determined by its associated Brauer configuration. Specifically, we can determine the indecomposable modules and the irreducible morphisms belonging to any component of the Auslander-Reiten quiver using only information from the Brauer configuration. We also show the number of certain components and their precise size and shape is entirely determined by the Green walks along the Brauer configuration. The second half of this thesis, comprising of the last two chapters, is a study on the representation type of symmetric special multiserial algebras. Unlike in the biserial case, not all of these algebras are tame. It is important to know if an algebra is tame or wild, since if it is wild, a classification of the indecomposable modules is considered to be hopeless. In this section of the thesis, we describe which symmetric special multiserial algebras are wild, which we present in terms of the Brauer configuration.