Planar harmonic and monogenic polynomials of type A

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 $1\leq j\leq N$,andtheLaplacian$\Delta_{\kappa}=\sum_{j=1}^{N}T_{j}^{2}$. 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.