Models for some irreducible representations of so (m, C) in discrete Clifford analysis

In this paper we work in the 'split' discrete Clifford analysis setting, i.e., the m-dimensional function theory concerning null-functions of the discrete Dirac operator partial derivative, defined on the grid Z(m), involving both forward and backward differences. This Dirac operator factorizes the (discrete) Star-Laplacian (Delta* = partial derivative(2)). We show how the space H-k of discrete k-homogeneous spherical harmonics, which is a reducible so(m, C)-representation, may explicitly be decomposed into 2(2m) isomorphic copies of irreducible so( m, C)representations with highest weight ( k, 0,..., 0). The key element is the introduction of 2(2m) idempotents, dividing the discrete Clifford algebra in 2(2m) subalgebras of dimension ((k) (k+m-1)) - ((k) (k+m-3)).