Planar harmonic and monogenic polynomials of type A

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 $1\le j\le N$, 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.

20 pages

References in corpus (1)

• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra