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. 2004). 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. 2004). 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 and zonal polynomials to Jack polynomials (Stanley 1989, Macdonald 1995). Jack polynomials are especially useful in the theory of random matrices (Dumitriu et al. 2004).

The Jack polynomials generalize the monomial scalar functions , which is 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. 2004).

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.

Letting  denote , the first few Jack "J" polynomials are given by

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

(Dumitriu et al. 2004).

Let  be a partition, then the Jack polynomials  can be defined as the functions that 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. 2004).

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

(8)

with eigenvalue  having highest-order term corresponding to  (Muirhead 1982, Dumitriu 2004). Here,

(9)

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

REFERENCES:

Dumitriu, I.; Edelman, A.; and Shuman, G. "MOPS: Multivariate Orthogonal Polynomials (Symbolically)." Preprint. March 26, 2004.

Foulkes, H. O. "A Survey of Some Combinatorial Aspects of Symmetric Functions." In Permutations. Paris: Gauthier-Villars, 1974.

Hua, L. K. Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Providence, RI: Amer. Math. Soc., 1963.

Jack, H. "A Class of Symmetric Polynomials with a Parameter." Proc. Roy. Soc. Edinburgh Sec. A: Math. Phys. Sci. 69, 1-18, 1969-70.

James, A. T. "The Distribution of the Latent Roots of the Covariance Matrix." Ann. Math. Stat. 31, 151-158, 1960.

James, A. T. "Distribution of Matrix Variates and Latent Roots Derived from Normal Samples." Ann. Math. Stat. 35, 475-501, 1964.

Kadell, K. "The Selberg-Jack Polynomials." Adv. Math. 130, 33-102, 1997.

Knop, F. and Sahi, S. "A Recursion and a Combinatorial Formula for the Jack Polynomials." Invent. Math. 128, 9-22, 1997.

Lasalle, M. "Some Combinatorial Conjectures for Jack Polynomials." Ann. Combin. 2, 61-83, 1998.

Macdonald, I. G. Symmetric Functions and Hall Polynomials, 2nd ed. Oxford, England: Oxford University Press, pp. 383 and 387, 1995.

Muirhead, R. J. Aspects of Multivariate Statistical Theory. New York: Wiley, 1982.

Stanley, R. P. "Some Combinatorial Properties of Jack Symmetric Functions." Adv. in Math. 77, 76-115, 1989.