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

SIGMA 11 (2015), 100, 21 pages      arXiv:1412.8001      https://doi.org/10.3842/SIGMA.2015.100
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Tableau Formulas for One-Row Macdonald Polynomials of Types $C_n$ and $D_n$

Boris Feigin a, Ayumu Hoshino b, Masatoshi Noumi c, Jun Shibahara d and Jun'ichi Shiraishi e
a) National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, Moscow, Russia
b) Kagawa National College of Technology, 355 Chokushi-cho, Takamatsu, Kagawa 761-8058, Japan
c) Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan
d) Hamamatsu University School of Medicine, 1-20-1 Handayama, Higashi-ku, Hamamatsu city, Shizuoka 431-3192, Japan
e) Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153-8914, Japan

Received May 19, 2015, in final form November 26, 2015; Published online December 05, 2015

Abstract
We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-row case. In view of the combinatorial structure, we call them ''tableau formulas''. For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ${}_{12}W_{11}$ series. We remark that the correlation functions of the deformed $\mathcal{W}$ algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.

Key words: Macdonald polynomials; deformed $\mathcal{W}$ algebras.

pdf (485 kb)   tex (24 kb)

References

1. Feigin B., Hoshino A., Shibahara J., Shiraishi J., Yanagida S., Kernel function and quantum algebras, in Representation Theory and Combinatorics, RIMS Kôky^uroku, Vol. 1689, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, 133-152, arXiv:1002.2485.
2. Frenkel E., Reshetikhin N., Deformations of ${\mathcal W}$-algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1998), 1-32, q-alg/9708006.
3. Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge, 1990.
4. Hoshino A., Noumi M., Shiraishi J., Some transformation formulas associated with Askey-Wilson polynomials and Lassalle's formulas for Macdonald-Koornwinder polynomials, arXiv:1406.1628.
5. Kashiwara M., Nakashima T., Crystal graphs for representations of the $q$-analogue of classical Lie algebras, J. Algebra 165 (1994), 295-345.
6. Koornwinder T.H., Askey-Wilson polynomials for root systems of type $BC$, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204.
7. Langer R., Schlosser M.J., Warnaar S.O., Theta functions, elliptic hypergeometric series, and Kawanaka's Macdonald polynomial conjecture, SIGMA 5 (2009), 055, 20 pages, arXiv:0905.4033.
8. Lassalle M., Some conjectures for Macdonald polynomials of type $B$, $C$, $D$, Sém. Lothar. Combin. 52 (2005), B52h, 24 pages, math.CO/0503149.
9. Lenart C., Haglund-Haiman-Loehr type formulas for Hall-Littlewood polynomials of type $B$ and $C$, Algebra Number Theory 4 (2010), 887-917, arXiv:0904.2407.
10. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
11. Macdonald I.G., Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000), B45a, 40 pages, math.QA/0011046.
12. Noumi M., Shiraishi J., A direct approach to the bispectral problem for the Ruijsenaars-Macdonald $q$-difference operators, arXiv:1206.5364.
13. Ram A., Yip M., A combinatorial formula for Macdonald polynomials, Adv. Math. 226 (2011), 309-331, arXiv:0803.1146.
14. van Diejen J.F., Emsiz E., Pieri formulas for Macdonald's spherical functions and polynomials, Math. Z. 269 (2011), 281-292, arXiv:1009.4482.