Jack function

Jack function

In mathematics, the Jack function, introduced by Henry Jack, is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials,and is in turn generalized by the Macdonald polynomials.

Definition

The Jack function J_kappa^{(alpha )}(x_1,x_2,ldots,x_m) of integer partition kappa, parameter alpha andarguments x_1,x_2,ldots, can be recursively defined as follows:

* For m=1 :

:: J_{(k)}^{(alpha )}(x_1)=x_1^k(1+alpha)cdots (1+(k-1)alpha)

* For m>1:

:: J_kappa^{(alpha )}(x_1,x_2,ldots,x_m)=sum_muJ_mu^{(alpha )}(x_1,x_2,ldots,x_{m-1})x_m^eta_{kappa mu},

: where the summation is over all partitions mu such that the skew partition kappa/mu is a horizontal strip, namely: kappa_1gemu_1gekappa_2gemu_2gecdotsgekappa_{n-1}gemu_{n-1}gekappa_n (mu_n must be zero or otherwise J_mu(x_1,ldots,x_{n-1})=0) and :: eta_{kappamu}=frac{ prod_{(i,j)in kappa} B_{kappamu}^kappa(i,j)}{prod_{(i,j)in mu} B_{kappamu}^mu(i,j)},

: where B_{kappamu}^ u(i,j) equals kappa_j'-i+alpha(kappa_i-j+1) if kappa_j'=mu_j' and kappa_j'-i+1+alpha(kappa_i-j) otherwise. The expressions kappa' and mu' refer to the conjugate partitions of kappa and mu, respectively. The notation (i,j)inkappa means that the product is taken over all coordinates (i,j) of boxes in the Young diagram of the partition kappa.

C normalization

The Jack functions form an orthogonal basis in a space of symmetric polynomials. This orthogonality property is unaffected by normalization. The normalization defined above is typically referred to as the J normalization. The C normalization is defined as:C_kappa^{(alpha)}(x_1,x_2,ldots,x_n)=frac{alpha^(|kappa|)!}{j_kappa}J_kappa^{(alpha)}(x_1,x_2,ldots,x_n),where:j_kappa=prod_{(i,j)in kappa}(kappa_j'-i+alpha(kappa_i-j+1))(kappa_j'-i+1+alpha(kappa_i-j)).

For alpha=2,; C_kappa^{(2)}(x_1,x_2,ldots,x_n) denoted often as justC_kappa(x_1,x_2,ldots,x_n) is known as the Zonal polynomial.

Connection with the Schur polynomial

When alpha=1 the Jack function is a scalar multiple of the Schur polynomial

:J^{(1)}_kappa(x_1,x_2,ldots,x_n) = H_kappa s_kappa(x_1,x_2,ldots,x_n),where:H_kappa=prod_{(i,j)inkappa} h_kappa(i,j)=prod_{(i,j)inkappa} (kappa_i+kappa_j'-i-j+1)is the product of all hook lengths of kappa.

Properties

If the partition has more parts than the number of variables, then the Jack function is 0:

:J_kappa^{(alpha )}(x_1,x_2,ldots,x_m)=0, mbox{ if }kappa_{m+1}>0.

Matrix argument

In some texts, especially in random matrix theory, authors have found it more convenient to use a matrix argument in the Jack function. The connection is simple. If X is a matrix with eigenvaluesx_1,x_2,ldots,x_m, then

:J_kappa^{(alpha )}(X)=J_kappa^{(alpha )}(x_1,x_2,ldots,x_m).

References

* James Demmel and Plamen Koev, "Accurate and efficient evaluation of Schur and Jack functions", "Math. Comp.", 75, no. 253, 223–239, 2006 (article electonically published August 31,2005)
* H. Jack, "A class of symmetric polynomials with a parameter", "Proc. Roy. Soc. Edinburgh Sect. A", 69, 1-18, 1970/1971.
*I. G. Macdonald, "Symmetric functions and Hall polynomials", Second ed., Oxford University Press, New York, 1995.
*Richard Stanley, "Some combinatorial properties of Jack symmetric functions", "Adv. Math.", 77, no. 1, 76–115, 1989.

External links

* [http://www-math.mit.edu/~plamen/software Software for computing the Jack function] by Plamen Koev.


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