Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

TL;DR

This paper studies polynomial spaces valued in dihedral group modules using Dunkl operators, explicitly finds the matrix weight function, and constructs an orthogonal basis with special hypergeometric coefficients.

Contribution

It provides an explicit form of the matrix weight function and constructs an orthogonal basis for harmonic polynomials in the dihedral group setting.

Findings

Explicit matrix weight function derived for the Gaussian form

Orthogonal basis for harmonic polynomials constructed

Polynomial coefficients expressed as balanced terminating $_4F_3$-series

Abstract

The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case for even dihedral groups). The matrix weight function for the Gaussian form is found explicitly by solving a boundary value problem, and then computing the normalizing constant. An orthogonal basis for the homogeneous harmonic polynomials is constructed. The coefficients of these polynomials are found to be balanced terminating $_4F_3$-series.