Generalised symmetries and bases for Dunkl monogenics

TL;DR

This paper introduces a new family of commuting symmetries for the Dunkl-Dirac operator, enabling the construction of polynomial bases for null-solutions and exploring their algebraic structure, especially for the a5a2a2 case.

Contribution

It presents a novel approach to constructing bases of Dunkl-Dirac null-solutions using generalized symmetries inspired by harmonic analysis.

Findings

Constructed bases of polynomial null-solutions for Dunkl-Dirac operator.

Identified the representation structure of these polynomial spaces.

Compared results with previous studies in the a5a2a2 case.

Abstract

We introduce a family of commuting generalised symmetries of the Dunkl--Dirac operator inspired by the Maxwell construction in harmonic analysis. We use these generalised symmetries to construct bases of the polynomial null-solutions of the Dunkl--Dirac operator. These polynomial spaces form representation spaces of the Dunkl--Dirac symmetry algebra. For the $\mathbb{Z}_2^d$ case, the results are compared with previous investigations.