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

TL;DR

This paper investigates the representation theory of fractional Brauer graph algebras of type MS, generalizing Brauer graph algebras, and characterizes their representation types and Auslander-Reiten components.

Contribution

It introduces Brauer G-sets, develops a covering theory for them, and characterizes the representation types of fractional Brauer graph algebras of type MS.

Findings

Characterization of representation types of fractional Brauer graph algebras of type MS.

Description of AR-components for representation-finite and domestic cases.

Development of a covering theory for Brauer G-sets.

Abstract

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.