Geodesic
Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: nonsymmetric Jack polynomials; Fourier–Stieltjes coefficients; matrix-valued measure; symmetric group modules. 
@article{SIGMA_2016_12_a32,
     author = {Charles F. Dunkl},
     title = {Orthogonality {Measure} on the {Torus} for {Vector-Valued} {Jack} {Polynomials}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a32/}
}
TY  - JOUR
AU  - Charles F. Dunkl
TI  - Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2016
VL  - 12
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a32/
LA  - en
ID  - SIGMA_2016_12_a32
ER  - 
%0 Journal Article
%A Charles F. Dunkl
%T Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
%J Symmetry, integrability and geometry: methods and applications
%D 2016
%V 12
%U http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a32/
%G en
%F SIGMA_2016_12_a32
Charles F. Dunkl. Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a32/

[1] Beerends R. J., Opdam E. M., “Certain hypergeometric series related to the root system $BC$”, Trans. Amer. Math. Soc., 339 (1993), 581–609 | DOI | MR | Zbl

[2] Dunkl C. F., “Differential-difference operators and monodromy representations of Hecke algebras”, Pacific J. Math., 159 (1993), 271–298 | DOI | MR | Zbl

[3] Dunkl C. F., “Symmetric and antisymmetric vector-valued Jack polynomials”, Sém. Lothar. Combin., 64 (2010), Art. B64a, 31 pp., arXiv: 1001.4485 | MR

[4] Dunkl C. F., “Vector polynomials and a matrix weight associated to dihedral groups”, SIGMA, 10 (2014), 044, 23 pp., arXiv: 1306.6599 | DOI | MR | Zbl

[5] Dunkl C. F., Luque J. G., “Vector-valued Jack polynomials from scratch”, SIGMA, 7 (2011), 026, 48 pp., arXiv: 1009.2366 | DOI | MR | Zbl

[6] Dunkl C. F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, 2nd ed., Cambridge University Press, Cambridge, 2014 | DOI | MR | Zbl

[7] Etingof P., Stoica E., “Unitary representations of rational Cherednik algebras”, Represent. Theory, 13 (2009), 349–370, arXiv: 0901.4595 | DOI | MR | Zbl

[8] Griffeth S., “Orthogonal functions generalizing Jack polynomials”, Trans. Amer. Math. Soc., 362 (2010), 6131–6157, arXiv: 0707.0251 | DOI | MR | Zbl

[9] James G., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing Co., Reading, Mass., 1981 | MR | Zbl

[10] Lapointe L., Vinet L., “Exact operator solution of the Calogero–Sutherland model”, Comm. Math. Phys., 178 (1996), 425–452 | DOI | MR | Zbl

[11] Murphy G. E., “A new construction of Young's seminormal representation of the symmetric groups”, J. Algebra, 69 (1981), 287–297 | DOI | MR | Zbl

[12] Opdam E. M., “Harmonic analysis for certain representations of graded Hecke algebras”, Acta Math., 175 (1995), 75–121 | DOI | MR | Zbl

[13] Rudin W., Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, 12, Interscience Publishers, New York–London, 1962 | MR | Zbl

[14] Vershik A. M., Okunkov A. Yu., J. Math. Sci., 131 (2005), A new approach to representation theory of symmetric groups, II, arXiv: math.RT/0503040 | DOI | MR