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Jack Polynomial

The Jack polynomials are a family of multivariate orthogonal polynomials dependent on a positive parameter alpha. Orthogonality of the Jack polynomials is proved in Macdonald (1995, p. 383). The Jack polynomials have a rich history, and special cases of alpha have been studied more extensively than others (Dumitriu et al. 2004). The following table summarizes some of these special cases.

alphaspecial polynomial
1/2quaternion zonal polynomial
1Schur polynomial
2zonal 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 alpha=1 were the Schur polynomials, and conjectured that alpha=2 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 x^k, which is orthogonal over the unit circle |z|=1 in the complex plane with weight function unity w(z)=1. The interval for the n-multivariate Jack polynomials I^n can therefore be thought of as an n-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 [1^n] equal to exactly n!, while the "P" normalization is monic.

Let m_alpha=m_alpha(X_1,...,X_n) denote the sum of all monomials x^beta where beta ranges over all distinct permutations of alpha=[alpha_1,...,alpha_n]. Then the first few Jack "J" polynomials are given by

J_([1])^alpha=m_([i])
(1)
J_([2])^alpha=(1+alpha)m_([2])+2m_([1,1])
(2)
J_([1,1])^alpha=2m_([1,1])
(3)
J_([3])^alpha=(1+alpha)(2+alpha)m_([3])+3(1+alpha)m_([2,1])+6m_([1,1,1])
(4)
J_([2,1])^alpha=(2+alpha)m_([2,1])+6m_([1,1,1])
(5)
J_([1,1,1])^alpha=6m_([1,1,1])
(6)

(Table 1 in Dumitriu et al. 2004).

Let lambda=[a_1,a_2,...,a_(l(lambda))] be a partition, then the Jack polynomials P_lambda^alpha can be defined as the functions that are orthogonal with respect to the inner product

<p_lambda,p_mu>_alpha=alpha^(l(lambda))z_lambdadelta_(lambdamu),
(7)

where delta_(ij) is the Kronecker delta and z_lambda=product_(i=1)^(l(lambda))a_1!i^(a_i), with a_i the number of occurrences of i in lambda (Macdonald 1995, Dumitriu et al. 2004).

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

D^*=sum_(i=1)^mx_i^2(d^2)/(dx_i^2)+2/alphasum_(1<=i!=j<=m)(x_i^2)/(x_i-x_j)d/(dx_i)
(8)

with eigenvalue rho_k^alpha+k(m-1) having highest-order term corresponding to kappa (Muirhead 1982, Dumitriu 2004). Here,

rho_kappa^alpha=sum_(i=1)^mk_i[k_i-1-2/alpha(i-1)]
(9)

and kappa is a partition of k and m is the number of variables.

See also

Macdonald Polynomial, Schur Polynomial, Zonal Polynomial

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References

Dumitriu, I.; Edelman, A.; and Shuman, G. "MOPS: Multivariate Orthogonal Polynomials (Symbolically)." https://arxiv.org/abs/math-ph/0409066. 24 Sep 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.

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Jack Polynomial

Cite this as:

Weisstein, Eric W. "Jack Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JackPolynomial.html

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