Macdonald polynomials

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In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal polynomials in several variables, introduced by Macdonald (1987). 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[edit]

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: \mu \le \lambda if and only if \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)_\infty = \prod_{r\ge0}(1-aq^r), the infinite q-Pochhammer symbol.
  • \Delta= \prod_{\alpha\in R} {(e^\alpha; q)_\infty \over (te^\alpha; q)_\infty}.
  • \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_\lambda=\sum_{\mu\le \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.

Examples[edit]

  • If q = t the Macdonald polynomials become the Weyl characters 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 the Hall–Littlewood polynomials when the root system has type A.
  • If t=1 the Macdonald polynomials become the sums over W orbits, which are the monomial symmetric functions when the root system has type A.
  • If we put t = qα and let q tend to 1 the Macdonald polynomials become Jack polynomials when the root system is of type A, and Heckman–Opdam polynomials for more general root systems.
  • For the affine root system A1, the Macdonald polynomials are the Rogers polynomials.
  • For the non-reduced rank 1 affine root system of type (C
    1
    , C1), the Macdonald polynomials are the Askey–Wilson polynomials, which in turn include as special cases most of the named families of orthogonal polynomials in 1 variable.
  • For the non-reduced affine root system of type (C
    n
    , Cn), the Macdonald polynomials are the

Koornwinder polynomials.

The Macdonald constant term conjecture[edit]

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

\langle P_\lambda, P_\lambda\rangle = \prod_{\alpha\in R, \alpha>0} \prod_{0<i<k} {1-q^{(\lambda+k\rho,2\alpha/(\alpha,\alpha))+i} \over 1-q^{(\lambda+k\rho,2\alpha/(\alpha,\alpha))-i}}.

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

\frac{P_\lambda(\dots,q^{\mu_i}t^{\rho_i},\dots)}{P_\lambda(t^\rho)}
=
\frac{P_\mu(\dots,q^{\lambda_i}t^{\rho_i},\dots)}{P_\mu(t^\rho)}.

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.

The Macdonald positivity conjecture[edit]

In the case of roots systems of type An−1 the Macdonald polynomials can be identified with symmetric polynomials in n variables (with coefficients that are rational functions of q and t). They can be related to Schur functions, and the coefficients Kλμ(q,t) of these relations 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[edit]

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})_{1\le i,j,\le n}

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

\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 Dμ. 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[edit]

In 2005, J. Haglund, M. Haiman and N. Loehr[1] gave the first proof of a combinatorial interpretation of the Macdonald polynomials, thus solving a 20-year old problem in algebraic combinatorics. The combinatorial formula immediately implies the integrality and positivity property of the Kostka-Macdonald coefficients Kλμ(q,t).

The formula is given as

P_\mu(x;q,t) = \sum_{\sigma:\mu \to \mathbb{Z}_+} q^{inv(\sigma)}t^{maj(\sigma)} x^{\sigma}

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 xσ should be interpreted as x_1^{\sigma_1} x_2^{\sigma_2} \cdots where σi is the number of boxes in the filling of μ with content i.

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