On hypergroups of matrices

Abstract

After recalling the definition and some basic properties of finite hypergroups-a notion introduced in a recent paper by one of the authors-several non-trivial examples of such hypergroups are constructed. The examples typically consist of n n×n matrices, each of which is an appropriate polynomial in a certain tri-diagonal matrix. The crucial result required in the construction is the following: 'let A be the matrix with ones on the super-and sub-diagonals, and with main diagonal given by a₁...a_n which are non-negative integers that form either a non-decreasing or a symmetric unimodal sequence; then A_k=P_k(A) is a non-negative matrix, where p_k denotes the characteristic polynomial of the top k×k principal submatrix of A, for k=1,...,n. The matrices A_k as well as the eigenvalues of A, are explicitly described in some special cases, such as (i) a_i=0 for all ior (ii) a_i=0 for i < n and a_n=1. Characters ot finite abelian hypergroups are defined, and that naturally leads to harmonic analysis on such hypergroups.