Irreducible actions of finite hypergroups and association schemes

The action of a finite hypergroup, introduced by Sunder and Wildberger [14], is an important tool in the study of finite hypergroups. It is defined via column-stochastic matrices. Finite hypergroups have close connections to association schemes. Actions of finite hypergroups have been used to construct association schemes in Sunder and Wildberger [14] and Xu [22]. In this paper, we study the characterizations of irreducible actions of finite hypergroups and investigate the constructions of association schemes arising from these irreducible actions. We will first obtain the characterizations of irreducible actions of finite hypergroups through the irreducibility of certain stochastic matrices. Then we establish some sufficient and necessary conditions under which an irreducible action of a finite hypergroup gives rise to an association scheme.