Planar harmonic and monogenic polynomials of type A

TL;DR

This paper investigates harmonic polynomials of type A associated with Dunkl operators, identifies those annihilated by specific operators, computes their structure constants, and constructs monogenic polynomials from them.

Contribution

It introduces a method to find harmonic polynomials annihilated by all but one Dunkl operator and constructs monogenic polynomials from these, advancing the understanding of Dunkl harmonic analysis.

Findings

Identified harmonic polynomials annihilated by all Dunkl operators except one.

Computed structure constants for these polynomials with respect to Gaussian and sphere inner products.

Constructed monogenic polynomials from the harmonic polynomials.

Abstract

Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on $\mathbb{R}^{N}$. The Dunkl operators are denoted by $T_{j}$ for $1\leq j\leq N$, and the Laplacian $\Delta_{\kappa}=\sum_{j=1}^{N}T_{j}^{2}$. This paper finds the homogeneous harmonic polynomials annihilated by all $T_{j}$ for $j>2$. The structure constants with respect to the Gaussian and sphere inner products are computed. These harmonic polynomials are used to produce monogenic polynomials, those annihilated by a Dirac-type operator.