Rotations in discrete Clifford analysis

Abstract

The Laplace and Dirac operators are rotation invariant operators which can be neatly expressed in (continuous) Euclidean Clifford analysis. In this paper, we consider the discrete counterparts of these operators, i.e. the discrete Laplacian Δ or star-Laplacian and the discrete Dirac operator ∂. We explicitly construct rotations operators for both of these differential operators (denoted by Ωa, b and dR(ea, b) respectively) in the discrete Clifford analysis setting. The operators Ωa, b satisfy the defining relations for so(m,C) and they are endomorphisms of the space Hk of k-homogeneous (discrete) harmonic polynomials, hence expressing Hk as a finite-dimensional so(m,C)-representation. Furthermore, the space Mk of (discrete) k-homogeneous monogenic polynomials can likewise be expressed as so(m,C)-representation by means of the operators dR(ea, b). We will also consider rotations of discrete harmonic (resp. monogenic) distributions, in particular point-distributions, which will allow us to evaluate functions in a rotated point. To make the discrete rotations more visual, we explicitly calculate the rotation of general point-distributions in two dimensions, showing the behavior of such discrete rotations in relation to the continuous case.