In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 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.
Definition First fix some notation:
R is a finite root system in a real vector space V. R+ is a choice of positive roots, to which corresponds a positive Weyl chamber. W is the Weyl group of R. Q is the root lattice of R (the lattice spanned by the roots). P is the weight lattice of R (containing Q). An ordering on the weights: μ ≤ λ {\displaystyle \mu \leq \lambda } if and only if λ − μ {\displaystyle \lambda -\mu } is a nonnegative linear combination of simple roots. P+ is the set of dominant weights: the elements of P in the positive Weyl chamber. ρ is the Weyl vector: half the sum of the positive roots; this is a special element of P+ in the interior of the positive Weyl chamber. F is a field of characteristic 0, usually the rational numbers. A = F(P) is the group algebra of P, with a basis of elements written eλ for λ ∈ P. If f = eλ, then f means e−λ, and this is extended by linearity to the whole group algebra. mμ = Σλ ∈ Wμeλ is an orbit sum; these elements form a basis for the subalgebra AW of elements fixed by W.
( a ; q ) ∞ = ∏ r ≥ 0 ( 1 − a q r ) {\displaystyle (a;q)_{\infty }=\prod _{r\geq 0}(1-aq^{r})} , the infinite q-Pochhammer symbol.
Δ = ∏ α ∈ R ( e α ; q ) ∞ ( t e α ; q ) ∞ . {\displaystyle \Delta =\prod _{\alpha \in R}{(e^{\alpha };q)_{\infty } \over (te^{\alpha };q)_{\infty }}.}
⟨ f , g ⟩ = ( constant term of f g ¯ Δ ) / | W | {\displaystyle \langle f,g\rangle =({\text{constant term of }}f{\overline {g}}\Delta )/|W|} is the inner product of two elements of A, at least when t is a positive integer power of q. The Macdonald polynomials Pλ for λ ∈ P+ are uniquely defined by the following two conditions:
P λ = ∑ μ ≤ λ u λ μ m μ {\displaystyle P_{\lambda }=\sum _{\mu \leq \lambda }u_{\lambda \mu }m_{\mu }} where uλμ is a rational function of q and t with uλλ = 1; Pλ and Pμ are orthogonal if λ < μ. 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. In the case of non-simply-laced root systems (B, C, F, G), the parameter t can be chosen to vary with the length of the root, giving a three-parameter family of Macdonald polynomials. One can also extend the definition to the nonreduced root system BC, in which case one obtains a six-parameter family (one t for each orbit of roots, plus q) known as Koornwinder polynomials. It is sometimes better to regard Macdonald polynomials as depending on a possibly non-reduced affine root system. In this case, there is one parameter t associated to each orbit of roots in the affine root system, plus one parameter q. The number of orbits of roots can vary from 1 to 5.
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