|
SIGMA 13 (2017), 040, 41 pages arXiv:1612.01486
https://doi.org/10.3842/SIGMA.2017.040
A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA
Received December 11, 2016, in final form June 02, 2017; Published online June 08, 2017
Abstract
For each irreducible module of the symmetric group $\mathcal{S}_{N}$ 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 two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the $N$-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper [SIGMA 12 (2016), 033, 27 pages] proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail. The $N$-torus is divided into $(N-1)!$ connected components by the hyperplanes $x_{i}=x_{j}$, $i$<$j$, which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.
Key words:
nonsymmetric Jack polynomials; matrix-valued weight function; symmetric group modules.
pdf (569 kb)
tex (45 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., Symmetric and antisymmetric vector-valued Jack polynomials, Sém. Lothar. Combin. 64 (2010), Art. B64a, 31 pages, arXiv:1001.4485.
-
Dunkl C.F., Orthogonality measure on the torus for vector-valued Jack polynomials, SIGMA 12 (2016), 033, 27 pages, arXiv:1511.06721.
-
Dunkl C.F., Vector-valued Jack polynomials and wavefunctions on the torus, J. Phys. A: Math. Theor. 50 (2017), 245201, 21 pages, arXiv:1702.02109.
-
Dunkl C.F., Luque J.G., Vector-valued Jack polynomials from scratch, SIGMA 7 (2011), 026, 48 pages, arXiv:1009.2366.
-
Felder G., Veselov A.P., Polynomial solutions of the Knizhnik-Zamolodchikov equations and Schur-Weyl duality, Int. Math. Res. Not. 2007 (2007), rnm046, 21 pages, math.RT/0610383.
-
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.
-
Opdam E.M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121.
-
Stembridge J.R., On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), 353-396.
-
Stoer J., Bulirsch R., Introduction to numerical analysis, Springer-Verlag, New York - Heidelberg, 1980.
|
|