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 Heckman–Opdam polynomials and Macdonald polynomials.

Definition

The Jack function $J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m)$ of integer partition $\kappa$, parameter $\alpha$ 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 $\mu$ such that the skew partition $\kappa/\mu$ is a horizontal strip, namely

$\kappa_1\ge\mu_1\ge\kappa_2\ge\mu_2\ge\cdots\ge\kappa_{n-1}\ge\mu_{n-1}\ge\kappa_n$ ($\mu_n$ 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 $\kappa_j'-i+\alpha(\kappa_i-j+1)$ 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 $(i,j)\in\kappa$ 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, with inner product: $\langle f,g\rangle = \int_{[0,2\pi]^n}f(e^{i\theta_1},\cdots,e^{i\theta_n})\overline{g(e^{i\theta_1},\cdots,e^{i\theta_n})}\prod_{1\le j

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.

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

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

is the product of all hook lengths of $\kappa$.

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 eigenvalues $x_1,x_2,\ldots,x_m$, then

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