Macdonald polynomials
In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal polynomials in several variables, introduced by Macdonald (1987). He later introduced a non-symmetric generalization in 1995. Macdonald originally associated his polynomials with weights λ of finite root systems and used just one variable t, but later realized that it is more natural to associate them with affine root systems rather than finite root systems, in which case the variable t can be replaced by several different variables t=(t1,...,tk), one for each of the k orbits of roots in the affine root system. The Macdonald polynomials are polynomials in n variables x=(x1,...,xn), where n is the rank of the affine root system. They generalize many other families of orthogonal polynomials, such as Jack polynomials and Hall–Littlewood polynomials and Askey–Wilson polynomials, which in turn include most of the named 1-variable orthogonal polynomials as special cases. Koornwinder polynomials are Macdonald polynomials of certain non-reduced root systems. They have deep relationships with affine Hecke algebras and Hilbert schemes, which were used to prove several conjectures made by Macdonald about them.
First fix some notation:
The Macdonald polynomials Pλ for λ ∈ P+ are uniquely defined by the following two conditions:
In other words, the Macdonald polynomials are obtained by orthogonalizing the obvious basis for AW. The existence of polynomials with these properties is easy to show (for any inner product). A key property of the Macdonald polynomials is that they are orthogonal: 〈Pλ, Pμ〉 = 0 whenever λ ≠ μ. This is not a trivial consequence of the definition because P+ is not totally ordered, and so has plenty of elements that are incomparable. Thus one must check that the corresponding polynomials are still orthogonal. The orthogonality can be proved by showing that the Macdonald polynomials are eigenvectors for an algebra of commuting self adjoint operators with 1-dimensional eigenspaces, and using the fact that eigenspaces for different eigenvalues must be orthogonal.
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