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.
Categories: math.CA