# The Spingroup and its actions in discrete Clifford analysis

## TL;DR

This paper introduces the discrete Spingroup, a double cover of SO(m), and demonstrates its action on discrete functions, establishing Spin(m)-representations and linking to Euclidean Clifford analysis.

## Contribution

It defines the discrete Spingroup and shows its action on discrete functions, connecting representation theory with practical calculations in discrete Clifford analysis.

## Key findings

- The discrete Spingroup acts on discrete functions.
- Spaces Hk and Mk are Spin(m)-representations.
- Results align with so(m, C) symmetry properties.

## Abstract

Recently, it has been established that the discrete star Laplace and the discrete Dirac operator, i.e. the discrete versions of their continuous counterparts when working on the standard grid, are rotation-invariant. This was done starting from the Lie algebra so(m, C) corresponding to the special orthogonal Lie group SO(m), considering its representation in the discrete Clifford algebra setting and proving that these operators are symmetries of the Dirac and Laplace operators. This set-up showed in an abstract way that representation-theoretically the discrete setting mirrors the Euclidean Clifford analysis setting. However from a practical point of view, the group-action remains indispensable for actual calculations. In this paper, we define the discrete Spingroup, which is a double cover of SO(m), and consider its actions on discrete functions. We show that this group-action makes the spaces Hk and Mk into Spin(m)-representations. We will often consider the compliance of our results to the results under the so(m, C)- action.