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If q = 0 the Macdonald polynomials become the (rescaled) zonal spherical functions for a semisimple p-adic group, or the Hall–Littlewood polynomials when the root system has type A. If t=1 the Macdonald polynomials become the sums over W orbits, which are the monomial symmetric functions when the root system has type A.
Using this he was able to give a proof of Macdonald's constant term conjecture for Macdonald polynomials (building on work of Eric Opdam). Another main inspiration for Cherednik to consider the double affine Hecke algebra was the quantum KZ equations .
A. A. Kirillov Lectures on affine Hecke algebras and Macdonald's conjectures Bull. Amer. Math. Soc. 34 (1997), 251–292. Macdonald, I. G. Affine Hecke algebras and orthogonal polynomials. Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, 2003. x+175 pp. ISBN 0-521-82472-9 MR 1976581
The Macdonald polynomials are a two-parameter family of orthogonal polynomials indexed by a positive weight λ of a root system, introduced by Ian G. Macdonald (1987). They generalize several other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials.
Macdonald reformulated these conjectures as conjectures about the norms of Macdonald polynomials. Macdonald's conjectures were proved by ( Cherednik 1995 ) using doubly affine Hecke algebras. Macdonald 's form of Dyson's conjecture for root systems of type BC is closely related to Selberg's integral .
In addition Jan Felipe van Diejen showed that the Macdonald polynomials associated to any classical root system can be expressed as limits or special cases of Macdonald-Koornwinder polynomials and found complete sets of concrete commuting difference operators diagonalized by them. [4] Furthermore, there is a large class of interesting families ...
In 2007, Haglund, Haiman and Loehr gave a combinatorial formula for the non-symmetric Macdonald polynomials. Haglund is the author of The q , t {\displaystyle q,t} -Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Combinatorics of Macdonald Polynomials .
SPOILERS BELOW—do not scroll any further if you don't want the answer revealed. The New York Times. Today's Wordle Answer for #1237 on Thursday, November 7, 2024.