Gabriel Diagrams Of Several Classes Of Finite EI Categories For The Symmetry Group S₄

Representation theory of groups has many applications in algebraic topol-ogy,algebraic geometry,geometric group theory,and algebraic combinatorics,and representation theory of EI categories is an important extension of group representation theory.Given a group,we can construct quite a few EI cate-gories such as transporter categories,orbit categories,Quillen categories,and fusion systems.These categories have played a crucial role in the represen-tation theory and homology theory of groups.In particular,we can obtain interesting representation theoretic properties of groups via describing struc-tures of these associated categories.Since the category algebra of a finite EI category,which is a finite dimen-sional associative algebra,can be represented by a quiver with relations,it is therefore helpful for us to construct the quivers with relations for EI categories associated to a given finite group,and use them to consider representations of this group.In this paper we describe the structure of the orbit category,transporter category,and Quillen category of the symmetric group S₄via explicitly com-puting the tables of morphisms,and construct the Gabriel quivers of the cat-egory algebras of the above mentioned categories.