Jack Polynomial

The Jack polynomials are a family of multivariate orthogonal polynomials dependent on a positive parameter . Orthogonality of the Jack polynomials is proved in Macdonald (1995, p. 383). The Jack polynomials have a rich history, and special cases of have been studied more extensively than others (Dumitriu et al. 2007). The following table summarizes some of these special cases.

	special polynomial
	quaternion zonal polynomial
1	Schur polynomial
2	zonal polynomial

Jack (1969-1970) originally defined the polynomials that eventually became associated with his name while attempting to evaluate an integral connected with the noncentral Wishart distribution (James 1960, Hua 1963, Dumitriu et al. 2007). Jack noted that the case were the Schur polynomials, and conjectured that were the zonal polynomials. The question of finding a combinatorial interpretation for the polynomials was raised by Foulkes (1974), and subsequently answered by Knop and Sahi (1997). Later authors then generalized many known properties of the Schur polynomials and zonal polynomials to Jack polynomials (Stanley 1989, Macdonald 1995). Jack polynomials are especially useful in the theory of random matrices (Dumitriu et al. 2007).

The Jack polynomials generalize the monomial scalar functions , which are orthogonal over the unit circle in the complex plane with weight function unity . The interval for the -multivariate Jack polynomials can therefore be thought of as an -dimensional torus (Dumitriu et al. 2007).

The Jack polynomials have several equivalent definitions (up to certain normalization constraints), and three common normalizations ("C," "J," and "P"). The "J" normalization makes the coefficient of the lowest-order monomial equal to exactly , while the "P" normalization is monic. The "C" normalization is characterized, for each nonnegative integer , by .

The multivariate Hermite polynomials, multivariate Jacobi polynomials, and multivariate Laguerre polynomials can be expanded in Jack polynomials. Jack polynomials also occur in the definition of the multivariate hypergeometric function (Dumitriu et al. 2007).

Let the monomial symmetric polynomial denote the sum of all monomials where ranges over all distinct permutations of . Then the first few Jack "J" polynomials are given by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

(Macdonald 1995, Dumitriu et al. 2007).

Let be a partition and , where is the th power sum. The monic Jack polynomials are triangular in the monomial symmetric polynomial basis with respect to dominance and are orthogonal with respect to the inner product

(7)

where is the Kronecker delta and , with the number of occurrences of in (Macdonald 1995, Dumitriu et al. 2007).

The Jack polynomial is a homogeneous polynomial eigenfunction of the Laplace-Beltrami operator-type differential operator

(8)

with eigenvalue . Its expansion in the symmetric monomial basis is triangular in dominance order, with leading term indexed by (Muirhead 1982, Dumitriu et al. 2007). Here,

(9)

and is a partition of and is the number of variables.

The trigonometric Calogero-Sutherland model gives another useful form of this differential operator,

(10)

It is related to by

(11)

Consequently, for padded with zeros, the monic Jack polynomial satisfies

(12)

Multiplying this polynomial by the trigonometric model's ground state produces an eigenfunction of its quantum Hamiltonian (Lapointe and Vinet 1996).