
SIGMA 4 (2008), 082, 9 pages arXiv:0812.0063
http://dx.doi.org/10.3842/SIGMA.2008.082
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 229044137, USA
Received October 14, 2008, in final form November 24, 2008; Published online November 29, 2008
Abstract
The symmetric group on 4 letters has the reflection group D_{3} as an
isomorphic image. This fact follows from the coincidence of the root systems
A_{3} and D_{3}. The isomorphism is used to construct an orthogonal basis
of polynomials of 4 variables with 2 parameters. There is an associated
quantum CalogeroSutherland model of 4 identical particles on the line.
Key words:
nonsymmetric Jack polynomials.
pdf (216 kb)
ps (167 kb)
tex (13 kb)
References
 Dunkl C.F., Singular polynomials and modules for the symmetric
groups, Int. Math. Res. Not. 2005 (2005), no. 39, 24092436, math.RT/0501494.
 Dunkl C.F., Xu Y., Orthogonal polynomials of
several variables, Encyclopedia of Mathematics and Its Applications,
Vol. 81, Cambridge University Press, Cambridge, 2001.
 Knop F., Sahi S., A recursion and a combinatorial formula
for Jack polynomials, Invent. Math. 128 (1997), 922, qalg/9610016.
 Lassalle M., Une formule de binôme généralisée
pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math.
310 (1990), 253256.
 Okounkov A., Olshanski G., Shifted Jack polynomials,
binomial formula, and applications, Math. Res. Lett. 4 (1997), 6978, qalg/9608020.

