Abstract
Harmonic polynomials of type A are polynomials annihilated by the Dunkl Laplacian associated to the symmetric group acting as a reflection group on R N . The Dunkl operators are denoted by T j for 1 ≤ j ≤ N , and the Laplacian Δ κ = ∑ 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.
Highlights
The symmetric group S N acts on x ∈ R N as a reflection group by permutation of coordinates
The corresponding measure on the N-torus is related to the Calogero–Sutherland quantum-mechanical model of N identical particles on the circle with 1/r2 interaction potential, and the measure wκ ( x ) e−| x| /2 dx is related to the model of N identical particles on the line with 1/r2 interactions and harmonic confinement (see [1] (Section 11.6))
This paper mainly concerns the measure on the unit sphere in R N for which there is an orthogonal decomposition involving harmonic polynomials
Summary
To start on the construction problem, we will determine all the harmonic homogeneous polynomials annihilated by Tj for 2 < j ≤ N. They are the analogues of ordinary harmonic polynomials in two variables and we call them planar. By means of Clifford algebra techniques, one can define an operator of Dirac type and Section 4 describes this theory and produces the planar monogenic polynomials.
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