Macdonald polynomial

In mathematics, Macdonald polynomials "P"_λ 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. 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 with a fixed Weyl chamber in a real vector space "V".
*"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")
*"P⁺" is the set of dominant weights: the elements of "P" in the Weyl chamber.
*ρ is the Weyl vector; the smallest element of "P"⁺ in the interior of the 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 "e"^λ for λ ∈ "P"
* If "f" = "e"^λ, then "f" means "e"^−λ, and this is extended by linearity to the whole group algebra.
*"m"_μ = Σ_λ∈"W"μ"e"^λ is an orbit sum; these elements form a basis for the subalgebra "A"^"W" of elements fixed by "W".
*$(a;q)\_infty\; =\; prod\_\{rge0\}(1-aq^r)$, a formal power series in "q".
*$Delta=\; prod\_\{alphain\; R\}\; \{(e^alpha;\; q)\_infty\; over\; (te^alpha;\; q)\_infty\}$
*The inner product 〈"f","g"〉 of two elements of "A" is defined to be:⟨"f","g"⟩ = (constant term of "f""g"Δ)/|W
at least when "t" is a positive integer power of "q".

The Macdonald polynomials "P"_λ for λ ∈ "P"⁺ are uniquely defined by the following two conditions::$P\_lambda=sum\_\{mule\; lambda\}u\_\{lambdamu\}m\_mu$ where "u"_λμ is a rational function of "q" and "t" with "u"_{λλ} = 1.: "P"_λ and "P"_μ are orthogonal if λ<μ

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, so has plenty of elements that are incomparable, and one has to check that the corresponding polynomials are still orthogonal. The orthgonality 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.

Examples

*If "q" = "t" the Macdonald polynomials become the Weyl character of the representations of the compact group of the root system, or the Schur functions in the case of root systems of type "A".
*If "q" = 0 the Macdonald polynomials become the (rescaled) zonal spherical functions for a semisimple "p"-adic group, or Hall–Littlewood polynomials when the roots system has type "A".
*If "t"=1 the Macdonald polynomials become the sums over "W" orbits, which are monomial symmetric functions when the root system has type "A".
*If we put "t" = "q"^α and let "q" tend to 1 the Macdonald polynomials become Jack polynomials when the root system is of type "A".
*If (1 − "t") = "k"(1 − "q") for some constant "k" and "q" is then set equal to 1 the Macdonald polynomials become the Jacobi polynomials "P"_λ("k") associated to a root system by Heckman and Opdam. For root systems of type "A" these are essentially the Jack polynomials mentioned above.

The Macdonald constant term conjecture

If "t" = "q"^"k" for some positive integer "k", then the norm of the Macdonald polynomials is given by

:

This was conjectured by Macdonald (1982), 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 "E"_"n" 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 "P"_λ at the point "t"^ρ, with ρ half the sum of the positive roots, and a symmetry

:$frac\{P\_lambda(dots,q^\{mu\_i\}t^\{\; ho\_i\},dots)\}\{P\_lambda(t^\; ho)\}=frac\{P\_mu(dots,q^\{lambda\_i\}t^\{\; ho\_i\},dots)\}\{P\_mu(t^\; ho)\}.$

Again, these were proved for general reduced root systems by Cherednik, 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 polynomialscan be identified with symmetric polynomials in "n" variables (with coefficients that are rational functions of "q" and "t"). They can be expanded in terms of Schur functions, and the coefficients "K"_λμ("q","t") of these expansions are called Kostka–Macdonald 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.

n! conjecture

The n! conjecture of Adriano Garsia and Mark Haiman states that for each partition μ of "n" the space

:$D\_mu\; =C\; [partial\; x,partial\; y]\; Delta\_mu$

spanned by all higher partial derivatives of

:$Delta\_mu\; =\; det\; (x\_i^\{p\_j\}y\_i^\{q\_j\})\_\{1le\; i,j,le\; n\}$

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:$Delta\_mu=x\_1y\_2+x\_2y\_3+x\_3y\_1-x\_2y\_1-x\_3y\_2-x\_1y\_3$:$y\_2-y\_3$:$y\_3-y\_1$:$x\_3-x\_2$:$x\_1-x\_3$:$1$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.

References

*Ivan Cherednik, [http://links.jstor.org/sici?sici=0003-486X%28199501%292%3A141%3A1%3C191%3ADAHAAM%3E2.0.CO%3B2-B "Double Affine Hecke Algebras and Macdonald's Conjectures"] The Annals of Mathematics > 2nd Ser., Vol. 141, No. 1 (Jan., 1995), pp. 191-216
*Adriano Garsia and Jeffrey B. Remmel [http://www.pnas.org/cgi/content/full/102/11/3891 "Breakthroughs in the theory of Macdonald polynomials"] PNAS March 15, 2005, vol. 102, no. 11, 3891–3894
*Mark Haiman [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cdm/1088530398 "Combinatorics, symmetric functions, and Hilbert schemes"] Current Developments in Mathematics 2002, no. 1 (2002), 39–111.
* Haiman, Mark [http://math.berkeley.edu/~mhaiman/ftp/newt-sf-2001/newt.pdf "Notes on Macdonald polynomials and the geometry of Hilbert schemes."] Symmetric functions 2001: surveys of developments and perspectives, 1–64, NATO Sci. Ser. II Math. Phys. Chem., 74, Kluwer Acad. Publ., Dordrecht, 2002.MathSciNet|id=2005f:14010
*Mark Haiman [http://arXiv.org/math.AG/0010246 "Hilbert schemes, polygraphs, and the Macdonald positivity conjecture"] J. Amer. Math. Soc. 14 (2001), 941–1006
*A. A. Kirillov [http://www.ams.org/bull/1997-34-03/S0273-0979-97-00727-1/home.html Lectures on affine Hecke algebras and Macdonald's conjectures] Bull. Amer. Math. Soc. 34 (1997), 251–292.
*Macdonald, I. G. "Some conjectures for root systems", SIAM J. Math. Anal. 13 (1982) 988–1007. DOI|10.1137/0513070
*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 DOI|10.2277/0521824729 MathSciNet|id=2005b:33021
*Macdonald, I. G. "Symmetric functions and Hall polynomials." Second edition. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN 0-19-853489-2 MathSciNet|id=96h:05207
*Macdonald, I. G. "Symmetric functions and orthogonal polynomials." Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, NJ. University Lecture Series, 12. American Mathematical Society, Providence, RI, 1998. xvi+53 pp. ISBN 0-8218-0770-6 MathSciNet|id=99f:05116
*Macdonald, I. G. "Affine Hecke algebras and orthogonal polynomials." Seminaire Bourbaki 797 (1995).
*Macdonald, I. G. [http://arxiv.org/abs/math.QA/0011046 "Orthogonal polynomials associated with root systems."] 1987 preprint, later published in Sém. Lothar. Combin. 45 (2000/01), Art. B45a, MathSciNet|id=2002a:33021

External links

*Garsia's page about [http://garsia.math.yorku.ca/MPWP/ Macdonald polynomials] .
*Some of [http://math.berkeley.edu/~mhaiman/ Haiman's papers] about Macdonald polynomials.