Jack function

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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.

1 Definition
2 C normalization
3 Connection with the Schur polynomial
4 Properties
5 Matrix argument
6 References
7 External link

[edit] Definition

The Jack function $J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m)$ of integer partition $κ$ , parameter $α$ and arguments $x_1,x_2,\ldots,$ 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 $m > 1$ :

$J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m)=\sum_\mu J_\mu^{(\alpha )}(x_1,x_2,\ldots,x_{m-1}) x_m^{|\kappa /\mu|}\beta_{\kappa \mu},$

where the summation is over all partitions

μ

such that the skew partition

κ / μ

is a horizontal strip, namely

$\kappa_1\ge\mu_1\ge\kappa_2\ge\mu_2\ge\cdots\ge\kappa_{n-1}\ge\mu_{n-1}\ge\kappa_n$ (

μ n

must be zero or otherwise $J_\mu(x_1,\ldots,x_{n-1})=0$ ) and

$\beta_{\kappa\mu}=\frac{ \prod_{(i,j)\in \kappa} B_{\kappa\mu}^\kappa(i,j) }{ \prod_{(i,j)\in \mu} B_{\kappa\mu}^\mu(i,j) },$

where $B_{\kappa\mu}^\nu(i,j)$ equals

κ j' - i + α(κ i - j + 1)

κ j' = μ j'

and

κ j' - i + 1 + α(κ i - j)

otherwise. The expressions

κ'

and

μ'

refer to the conjugate partitions of

κ

and

μ

, respectively. The notation $(i,j)\in\kappa$ means that the product is taken over all coordinates

(i, j)

of boxes in the Young diagram of the partition

κ

[edit] 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

$C_\kappa^{(\alpha)}(x_1,x_2,\ldots,x_n) = \frac{\alpha^{|\kappa|}(|\kappa|)!} {j_\kappa} J_\kappa^{(\alpha)}(x_1,x_2,\ldots,x_n),$

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 just $C_\kappa(x_1,x_2,\ldots,x_n)$ is known as the Zonal polynomial.

[edit] Connection with the Schur polynomial

When $α = 1$ 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

$H_\kappa=\prod_{(i,j)\in\kappa} h_\kappa(i,j)= \prod_{(i,j)\in\kappa} (\kappa_i+\kappa_j'-i-j+1)$

is the product of all hook lengths of $κ$ .

[edit] Properties

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

$J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m)=0, \mbox{ if }\kappa_{m+1}>0.$

[edit] 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 eigenvalues $x_1,x_2,\ldots,x_m$ , then

$J_\kappa^{(\alpha )}(X)=J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m).$

[edit] References

James Demmel and Plamen Koev, "Accurate and efficient evaluation of Schur and Jack functions", Math. Comp., 75, no. 253, 223–239, 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.