Finite Hypergroups and Their Representation Theory

Abstract

A hypergroup is an algebraic structure generalizing the concept of a group. This is done by adding a sense in which the multiplication can be interpreted as probabilistic by letting the operation range over an algebra. This leads to weakening the requirement of invertibility that we have for ordinary groups. While hypergroups have been studied in full generality, finite hypergroups are an interesting special case which can be dealt with by more elementary methods. In this report we restrict ourselves to finite hypergroups and develop the representation theory of hypergroups by trying to generalize the well-known representation theory of finite groups. While many proofs transfer immediately, some proofs that depend on the invertibility of group elements must be modified. We prove the Schur orthogonality relations for hypergroup representations, and establish character orthogonality. Finally, we restrict ourselves to commutative finite hypergroups and prove some interesting results about such objects. This naturally leads to the development of Fourier analysis on finite hypergroups, using similar techniques as in the finite group case. In the appendices we consider several examples of hypergroups coming from finite groups and distance-transitive graphs. All of these hypergroups are commutative, and therefore the entire body of results apply to them.