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

SIGMA 4 (2008), 082, 9 pages      arXiv:0812.0063
Contribution to the Special Issue on Dunkl Operators and Related Topics

Some Orthogonal Polynomials in Four Variables

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

Received October 14, 2008, in final form November 24, 2008; Published online November 29, 2008

The symmetric group on 4 letters has the reflection group D3 as an isomorphic image. This fact follows from the coincidence of the root systems A3 and D3. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical particles on the line.

Key words: nonsymmetric Jack polynomials.

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  1. Dunkl C.F., Singular polynomials and modules for the symmetric groups, Int. Math. Res. Not. 2005 (2005), no. 39, 2409-2436, math.RT/0501494.
  2. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  3. Knop F., Sahi S., A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 9-22, q-alg/9610016.
  4. Lassalle M., Une formule de binôme généralisée pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 253-256.
  5. Okounkov A., Olshanski G., Shifted Jack polynomials, binomial formula, and applications, Math. Res. Lett. 4 (1997), 69-78, q-alg/9608020.

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