Macdonald polynomials

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.

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.

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.

It is sometimes better to regard Macdonald polynomials as depending on a possibly non-reduced affine root system.

If t = qk for some positive integer k, then the norm of the Macdonald polynomials is given by This was conjectured by Macdonald (1982) as a generalization of the Dyson conjecture, and proved for all (reduced) root systems by Cherednik (1995) using properties of double affine Hecke algebras.

The conjecture had previously been proved case-by-case for all roots systems except those of type En by several authors.

There are two other conjectures which together with the norm conjecture are collectively referred to as the Macdonald conjectures in this context: in addition to the formula for the norm, Macdonald conjectured a formula for the value of Pλ at the point tρ, and a symmetry Again, these were proved for general reduced root systems by Cherednik (1995), using double affine Hecke algebras, with the extension to the BC case following shortly thereafter via work of van Diejen, Noumi, and Sahi.

In the case of roots systems of type An−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

of the Macdonald polynomials (see Combinatorial formula below) form an orthogonal basis of the space of symmetric functions over

It is still a central open problem in algebraic combinatorics to find a combinatorial formula for the qt-Kostka coefficients.

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 (pj, qj) 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 (pj, qj) are (0, 0), (0, 1), (1, 0), and the space Dμ is spanned by which has dimension 6 = 3!.

conjecture involved showing that the isospectral Hilbert scheme of n points in a plane was Cohen–Macaulay (and even Gorenstein).

conjecture implied that the Kostka–Macdonald coefficients were graded character multiplicities for the modules Dμ.

This immediately implies the Macdonald positivity conjecture because character multiplicities have to be non-negative integers.

In 2005, J. Haglund, M. Haiman and N. Loehr[1] gave the first proof of a combinatorial interpretation of the Macdonald polynomials.

Macdonald’s formula is different to that in Haglund, Haiman, and Loehr's work, with many fewer terms (this formula is proved also in Macdonald's seminal work,[3] Ch.

While very useful for computation and interesting in its own right, their combinatorial formulas do not immediately imply positivity of the Kostka-Macdonald coefficients

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 xσ should be interpreted as

French notation is more commonly used in the study of Macdonald polynomials.

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.

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.

[4] 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.

is a weighting function mapping those queues to specific polynomials.

The symmetric Macdonald polynomial satisfies: where the outer sum is over all distinct compositions