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


SIGMA 3 (2007), 052, 19 pages      math.CA/0702107      https://doi.org/10.3842/SIGMA.2007.052

Polynomials Associated with Dihedral Groups

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

Received February 06, 2007; Published online March 22, 2007

Abstract
There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The last section of the paper deals with parameter values for which the formulae have singularities.

Key words: intertwining operator; Jacobi polynomials.

pdf (285 kb)   ps (212 kb)   tex (21 kb)

References

  1. Berenstein A., Burman Y., Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras, math.RT/0505173.
  2. Dunkl C., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  3. Dunkl C., Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry, J. Math. Anal. Appl. 143 (1989), 459-470.
  4. Dunkl C., Operators commuting with Coxeter group actions on polynomials, in Invariant Theory and Tableaux, Editor D. Stanton, Springer, Berlin - Heidelberg - New York, 1990, 107-117.
  5. Dunkl C., Integral kernels with reflection group invariance, Can. J. Math. 43 (1991), 1213-1227.
  6. Dunkl C., de Jeu M., Opdam E., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256.
  7. Dunkl C., Xu Y., Orthogonal polynomials of several variables, Encycl. of Math. and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  8. Rösler M., Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463, q-alg/9710029.
  9. Scalas F., Poisson integrals associated to Dunkl operators for dihedral groups, Proc. Amer. Math. Soc., 133 (2005), 1713-1720.
  10. Xu Y., Intertwining operator and h-harmonics associated with reflection groups, Can. J. Math. 50 (1998), 193-208.


Previous article   Next article   Contents of Volume 3 (2007)