Fractional Brauer configuration algebras III: fractional Brauer graph algebras of type MS

In previous two papers, we defined fractional Brauer configuration algebras and developed their covering theory. In this paper, we study the representation theory of fractional Brauer graph algebras of type MS, a special class of fractional Brauer configuration algebras that properly generalizes Brauer graph algebras. We first introduce the notion of Brauer $G$-set, which is a generalization of fractional Brauer graph of type MS. Then we develop a covering theory for Brauer $G$-sets and use it to characterize the representation types of fractional Brauer graph algebras of type MS. Moreover, we describe the AR-components of representation-finite and domestic fractional Brauer graph algebras of type MS respectively.