# Spinorspaces in discrete Clifford analysis

## TL;DR

This paper explores the structure of discrete spherical monogenics in split discrete Clifford analysis, revealing their decomposition into irreducible representations with specific highest weights in both odd and even dimensions.

## Contribution

It provides a detailed decomposition of the space of discrete homogeneous spherical monogenics into irreducible representations, highlighting differences between odd and even dimensions.

## Key findings

- Decomposition of M_k into irreducible representations with specific highest weights.
- Explicit description of representation structure in odd and even dimensions.
- Advancement in understanding discrete Clifford analysis and monogenic functions.

## Abstract

In this paper we work in the `split' discrete Clifford analysis setting, i.e. the m-dimensional function theory concerning null-functions, defined on the grid Z^m, of the discrete Dirac operator D, involving both forward and backward differences, which factorizes the (discrete) Star-Laplacian (Delta = D^2). We show how the space M_k of discrete homogeneous spherical monogenics of degree k, is decomposable into 2^{2m-n} isomorphic irreducible representations with highest weight (k + 1/2, 1/2,...,1/2) in the odd-dimensional case and two times 2^{2m-n} isomorphic irreducible representations with highest weight (k)'_+ = (k + 1/2, 1/2,...,1/2,1/2) resp. (k)'_- = (k + 1/2, 1/2,...,1/2,-1/2) in the even dimensional case.