Models for Some Irreducible Representations of $$\mathfrak{so}(m, \mathbb{C})$$ in Discrete Clifford Analysis

Abstract

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 ∂, defined on the grid \( \mathbb{Z} \) m , involving both forward and backward differences. This Dirac operator factorizes the (discrete) Star- Laplacian (Δ* = ∂2). We show how the space \( \mathcal{H}_{k} \) of discrete k-homogeneous spherical harmonics, which is a reducible \(\mathfrak{so} (m, \mathbb{C})\)-representation, may explicitly be decomposed into 22m isomorphic copies of irreducible \(\mathfrak{so} (m, \mathbb{C})\)- representations with highest weight (k, 0, …, 0). The key element is the introduction of 22m idempotents, dividing the discrete Clifford algebra in 22m subalgebras of dimension \( \left(\frac{k+m-1}{k}\right) - \left(\frac{k+m-3}{k}\right) \).