|
SIGMA 12 (2016), 033, 27 pages arXiv:1511.06721
https://doi.org/10.3842/SIGMA.2016.033
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications
Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA
Received November 26, 2015, in final form March 23, 2016; Published online March 27, 2016
Abstract
For each irreducible module of the symmetric group on $N$ objects there is a set of parametrized nonsymmetric Jack polynomials in $N$ variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to certain Hermitian forms. These polynomials were studied by the author and J.-G. Luque using a Yang-Baxter graph technique. This paper constructs a matrix-valued measure on the $N$-torus for which the polynomials are mutually orthogonal. The construction uses Fourier analysis techniques. Recursion relations for the Fourier-Stieltjes coefficients of the measure are established, and used to identify parameter values for which the construction fails. It is shown that the absolutely continuous part of the measure satisfies a first-order system of differential equations.
Key words:
nonsymmetric Jack polynomials; Fourier-Stieltjes coefficients; matrix-valued measure; symmetric group modules.
pdf (480 kb)
tex (34 kb)
References
-
Beerends R.J., Opdam E.M., Certain hypergeometric series related to the root system $BC$, Trans. Amer. Math. Soc. 339 (1993), 581-609.
-
Dunkl C.F., Differential-difference operators and monodromy representations of Hecke algebras, Pacific J. Math. 159 (1993), 271-298.
-
Dunkl C.F., Symmetric and antisymmetric vector-valued Jack polynomials, Sém. Lothar. Combin. 64 (2010), Art. B64a, 31 pages, arXiv:1001.4485.
-
Dunkl C.F., Vector polynomials and a matrix weight associated to dihedral groups, SIGMA 10 (2014), 044, 23 pages, arXiv:1306.6599.
-
Dunkl C.F., Luque J.G., Vector-valued Jack polynomials from scratch, SIGMA 7 (2011), 026, 48 pages, arXiv:1009.2366.
-
Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014.
-
Etingof P., Stoica E., Unitary representations of rational Cherednik algebras, Represent. Theory 13 (2009), 349-370, arXiv:0901.4595.
-
Griffeth S., Orthogonal functions generalizing Jack polynomials, Trans. Amer. Math. Soc. 362 (2010), 6131-6157, arXiv:0707.0251.
-
James G., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.
-
Lapointe L., Vinet L., Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), 425-452.
-
Murphy G.E., A new construction of Young's seminormal representation of the symmetric groups, J. Algebra 69 (1981), 287-297.
-
Opdam E.M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121.
-
Rudin W., Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, Vol. 12, Interscience Publishers, New York - London, 1962.
-
Vershik A.M., Okunkov A.Yu., A new approach to representation theory of symmetric groups. II, J. Math. Sci. 131 (2005), 5471-5494, math.RT/0503040.
|
|