Macdonald polynomial

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In mathematics, Macdonald polynomials Pλ 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.

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[edit] 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 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), a formal power series in q.
  • \Delta= \prod_{\alpha\in 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 fgΔ)/|W|

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, 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.

[edit] 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 = qα 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 Pλ(k) associated to a root system by Heckman and Opdam. For root systems of type A these are essentially the Jack polynomials mentioned above.

[edit] The Macdonald constant term conjecture

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), and proved for all 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.

[edit] The Macdonald positivity conjecture

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 expanded in terms of Schur functions, and the coefficients Kλμ(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).

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

Δμ = x1y2 + x2y3 + x3y1 − x2y1 − x3y2 − x1y3
y2 − y3
y3 − y1
x3 − x2
x1 − x3
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.

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