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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.
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.
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.
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
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 .
Despite the greatest strides in mathematics, these hard math problems remain unsolved. Take a crack at them yourself. ... For example, x²-6 is a polynomial with integer coefficients, since 1 and ...
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 ...
The problem of determining if a given set of Wang tiles can tile the plane. The problem of determining the Kolmogorov complexity of a string. Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers.