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

SIGMA 16 (2020), 118, 11 pages      arXiv:2004.12875      https://doi.org/10.3842/SIGMA.2020.118
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

New Pieri Type Formulas for Jack Polynomials and their Applications to Interpolation Jack Polynomials

Genki Shibukawa
Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

Received April 29, 2020, in final form November 14, 2020; Published online November 21, 2020

Abstract
We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by Knop and Sahi. We also propose Pieri formulas for interpolation Jack polynomials and intertwining relations for a kernel function for Jack polynomials.

Key words: Jack polynomial; interpolation Jack polynomial; Pieri formula; kernel function.

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References

1. Coskun H., Gustafson R.A., Well-poised Macdonald functions $W_\lambda$ and Jackson coefficients $\omega_\lambda$ on $BC_n$, in Jack, Hall-Littlewood and Macdonald polynomials, Contemp. Math., Vol. 417, Amer. Math. Soc., Providence, RI, 2006, 127-155, arXiv:math.CO/0412153.
2. Debiard A., Polynômes de Tchébychev et de Jacobi dans un espace euclidien de dimension $p$, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 529-532.
3. Knop F., Sahi S., Difference equations and symmetric polynomials defined by their zeros, Int. Math. Res. Not. 1996 (1996), 473-486, arXiv:q-alg/9610017.
4. Koornwinder T.H., Okounkov's $BC$-type interpolation Macdonald polynomials and their $q=1$ limit, Sém. Lothar. Combin. 72 (2015), Art. B72a, 27 pages, arXiv:1408.5993.
5. Lassalle M., Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), 289-324.
6. Macdonald I.G., Schur functions: theme and variations, in Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), Publ. Inst. Rech. Math. Av., Vol. 498, University Louis Pasteur, Strasbourg, 1992, 5-39.
7. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
8. Okounkov A., Binomial formula for Macdonald polynomials and applications, Math. Res. Lett. 4 (1997), 533-553, arXiv:q-alg/9608021.
9. Okounkov A., ${\rm BC}$-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998), 181-207, arXiv:q-alg/9611011.
10. Okounkov A., Olshanski G., Shifted Jack polynomials, binomial formula, and applications, Math. Res. Lett. 4 (1997), 69-78, arXiv:q-alg/9608020.
11. Okounkov A., Olshanski G., Shifted Schur functions, St. Petersburg Math. J. 9 (1998), 239-300, arXiv:q-alg/9605042.
12. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, arXiv:math.CO/0402113.
13. Sahi S., The spectrum of certain invariant differential operators associated to a Hermitian symmetric space, in Lie Theory and Geometry, Progr. Math., Vol. 123, Birkhäuser Boston, Boston, MA, 1994, 569-576.
14. Sahi S., Interpolation, integrality, and a generalization of Macdonald's polynomials, Int. Math. Res. Not. 1996 (1996), 457-471.
15. Sekiguchi J., Zonal spherical functions on some symmetric spaces, Publ. Res. Inst. Math. Sci. 12 (1977), 455-459.
16. Stanley R.P., Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76-115.
17. Vilenkin N.Ja., Klimyk A.U., Representation of Lie groups and special functions. Recent advances, Mathematics and its Applications, Vol. 316, Kluwer Academic Publishers Group, Dordrecht, 1995.