Macdonald polynomials

In mathematics, Macdonald polynomials "P"λ("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"=("t"1...,"t""k"), one for each of the "k" orbits of roots in the affine root system. The Macdonald polynomials are polynomials in "n" variables "x"=("x"1...,"x""n"), 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:
The Macdonald polynomials "P"λ 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 "A""W". 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: 〈"P"λ, "P"μ〉 = 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.
The Macdonald constant term conjecture.
If "t" = "q""k" for some positive integer "k", then the norm of the Macdonald polynomials is given by
Again, these were proved for general reduced root systems by , using double affine Hecke algebras, with the extension to the BC case following shortly thereafter via work of van Diejen, Noumi, and Sahi.
The Macdonald positivity conjecture.
In the case of roots systems of type "A""n"−1 the Macdonald polynomials
are simply symmetric polynomials in "n" variables with coefficients that are rational functions of "q" and "t". A certain transformed version formula_8 of the Macdonald polynomials (see Combinatorial formula below) form an orthogonal basis of the space of symmetric functions over formula_9, and therefore can be expressed in terms of Schur functions formula_10. The coefficients "K"λμ("q","t") of these relations are called Kostka–Macdonald coefficients or "qt"-Kostka coefficients.
Macdonald conjectured that the Kostka–Macdonald coefficients were polynomials in "q" and "t" with non-negative integer coefficients. These conjectures are now proved; the hardest and final step was proving the positivity, which was done by Mark Haiman (2001), by proving the "n"! conjecture.
It is still a central open problem in algebraic combinatorics to find a combinatorial formula for the "qt"-Kostka coefficients.
n! conjecture.
The "n"! conjecture of Adriano Garsia and Mark Haiman states that for each partition μ of "n" the space
spanned by all higher partial derivatives of
has dimension "n"!, where ("p""j", "q""j") run through the "n" elements of the diagram of the partition μ, regarded as a subset of the pairs of non-negative integers. 
For example, if μ is the partition 3 = 2 + 1 of "n" = 3 then the pairs ("p""j", "q""j") are
(0, 0), (0, 1), (1, 0), and the space "D"μ is spanned by
which has dimension 6 = 3!.
Haiman's proof of the Macdonald positivity conjecture and the "n"! conjecture involved showing that the isospectral Hilbert scheme of "n" points in a plane was Cohen–Macaulay (and even Gorenstein). Earlier results of Haiman and Garsia had already shown that this implied the "n"! conjecture, and that the "n"! conjecture implied that the Kostka–Macdonald coefficients were graded character multiplicities for the modules "D"μ. This immediately implies the Macdonald positivity conjecture because character multiplicities have to be non-negative integers.
Ian Grojnowski and Mark Haiman found another proof of the Macdonald positivity conjecture by proving a positivity conjecture for LLT polynomials.
Combinatorial formula for the Macdonald polynomials.
In 2005, J. Haglund, M. Haiman and N. Loehr gave the first proof of a combinatorial interpretation of the 
Macdonald polynomials. While very useful for computation and interesting in its own right, this combinatorial formula does not immediately imply positivity of the Kostka-Macdonald coefficients formula_19 as it gives the decomposition of the Macdonald polynomials into monomial symmetric functions rather than into Schur functions.
The formula, which involves the "transformed Macdonald polynomials" formula_8 rather than the usual formula_21, is given as
where σ is a filling of the Young diagram of shape μ, "inv" and "maj" are certain combinatorial statistics (functions) defined on the filling σ. This formula expresses the Macdonald polynomials in infinitely many variables. To obtain the polynomials in "n" variables, simply restrict the formula to fillings that only use the integers 1, 2, ..., "n". The term "x"σ should be interpreted as formula_23 where "σi" is the number of boxes in the filling of μ with content "i".
The transformed Macdonald polynomials formula_24 in the formula above are related to the classical Macdonald polynomials formula_25 via a sequence of transformations. First, the "integral form" of the Macdonald polynomials, denoted formula_26, is a re-scaling of formula_27 that clears the denominators of the coefficients:
where formula_29 is the collection of squares in the Young diagram of formula_30, and formula_31 and formula_32 denote the "arm" and "leg" of the square formula_33, as shown in the figure. "Note: The figure at right uses French notation for tableau, which is flipped vertically from the English notation used on the Wikipedia page for Young diagrams. French notation is more commonly used in the study of Macdonald polynomials."
The transformed Macdonald polynomials formula_24 can then be defined in terms of the formula_35's. We have
where 
The bracket notation above denotes plethystic substitution.
This formula can be used to prove Knop and Sahi's formula for the Jack polynomials.
Non-symmetric Macdonald polynomials.
In 1995, Macdonald introduced a non-symmetric analogue of the symmetric Macdonald polynomials,
and the symmetric Macdonald polynomials can easily be recovered from the non-symmetric counterpart.
In his original definition, he shows that the non-symmetric Macdonald polynomials are a unique family of 
polynomials orthogonal to a certain inner product, as well as satisfying a 
triangularity property when expanded in the monomial basis.
In 2007, Haglund, Haiman and Loehr gave a combinatorial formula for the non-symmetric Macdonald polynomials.
The non-symmetric Macdonald polynomials specialize to Demazure characters by taking q=t=0,
and to key polynomials when q=t=∞.
Combinatorial formulae based on the exclusion process.
In 2018, S. Corteel, O. Mandelshtam, and L. Williams used the exclusion process to give a direct combinatorial characterization of both symmetric and nonsymmetric Macdonald polynomials. Their results differ from the earlier work of Haglund in part because they give a formula directly for the Macdonald polynomials rather than a transformation thereof. They develop the concept of a multiline queue, which is a matrix containing balls or empty cells together with a mapping between balls and their neighbors and a combinatorial labeling mechanism. The nonsymmetric Macdonald polynomial then satisfies:
where the sum is over all formula_39 multiline queues of type formula_30 and formula_41 is a weighting function mapping those queues to specific polynomials. The symmetric Macdonald polynomial satisfies:
where the outer sum is over all distinct compositions formula_43 which are permutations of formula_30, and the inner sum is as before.