Some Properties of Matrix Harmonics on S 2

Abstract

We show that matrix harmonics on S2 (obtained from harmonic polynomials in 3 variables by replacing the commuting variables x1, x2, x3 by hermitian N×N matrices X1, X2, X3 satisfying \(\), + cycl.) define two sets of families of discrete orthogonal polynomials, dual to each other, one of them having 3-term recurrence relations that, written in tridiagonal matrix form, are the constituents of a discrete Laplacian whose eigenvalues coincide with the first N2 ones of the ordinary Laplacian on S2.