Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 044, 23 pages      arXiv:1306.6599

Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA

Received January 22, 2014, in final form April 10, 2014; Published online April 15, 2014

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 4F3-series.

Key words: standard module; Gaussian weight.

pdf (398 kb)   tex (24 kb)


  1. Dunkl C.F., Differential-difference operators and monodromy representations of Hecke algebras, Pacific J. Math. 159 (1993), 271-298.
  2. Dunkl C.F., Vector-valued polynomials and a matrix weight function with B2-action, SIGMA 9 (2013), 007, 23 pages, arXiv:1210.1177.
  3. Dunkl C.F., Dunkl operators and related special functions, arXiv:1210.3010.
  4. Etingof P., Stoica E., Unitary representations of rational Cherednik algebras, Represent. Theory 13 (2009), 349-370, arXiv:0901.4595.
  5. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010.

Previous article  Next article   Contents of Volume 10 (2014)