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 $\kappa$.

Combinatorial formula

In 1997, F. Knop and S. Sahi gave a purely combinatorial formula for the Jack polynomials $J_\mu^{(\alpha )}$ in n variables:

$J_\mu^{(\alpha )} = \sum_{T} d_T(\alpha) \prod_{s \in T} x_{T(s)}$.

The sum is taken over all admissible tableaux of shape $\lambda$, and $d_T(\alpha) = \prod_{s \in T \text{ critical}} d_\lambda(\alpha)(s)$ with $d_\lambda(\alpha)(s) = \alpha(a_\lambda(s) +1) + (l_\lambda(s) + 1)$.

An admissible tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with numbers 1,2,…,n such that for any box (i,j) in the tableau,

• T(i,j) ≠ T( i',j) whenever i' > i.
• T(i,j) ≠ T( i',j-1) whenever j>1 and i' < i.

A box $s = (i,j) \in \lambda$ is critical for the tableau T if j>1 and $T(i,j)=T(i,j-1)$.

This result can be seen as a special case of the more general combinatorial formula for Macdonald polynomials.

C normalization

The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product: $\langle f,g\rangle = \int_{[0,2\pi]^n}f(e^{i\theta_1},\cdots,e^{i\theta_n})\overline{g(e^{i\theta_1},\cdots,e^{i\theta_n})}\prod_{1\le j<k\le n}|e^{i\theta_j}-e^{i\theta_k}|^{2/\alpha}d\theta_1\cdots d\theta_n$

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.

P normalization

The P normalization is given by the identity $J_\lambda = H'_\lambda P_\lambda$, where $H'_\lambda = \prod_{s\in \lambda} (\alpha a_\lambda(s) + l_\lambda(s) + 1)$ and $a_\lambda$ and $l_\lambda$ denotes the arm and leg length respectively. Therefore, for $\alpha=1$, $P_\lambda$ is the usual Schur function.

Similar to Schur polynomials, $P_\lambda$ can be expressed as a sum over Young tableaux. However, one need to add an extra weight to each tableau that depends on the parameter $\alpha$.

Thus, a formula for the Jack function $P_\lambda$ is given by

$P_\lambda = \sum_{T} \psi_T(\alpha) \prod_{s \in \lambda} x_{T(s)}$

where the sum is taken over all tableaux of shape $\lambda$, and $T(s)$ denotes the entry in box s of T.

The weight $\psi_T(\alpha)$ can be defined in the following fashion: Each tableau T of shape $\lambda$ can be interpreted as a sequence of partitions $\emptyset = \nu_1 \to \nu_2 \to \dots \to \nu_n = \lambda$ where $\nu_{i+1}/\nu_i$ defines the skew shape with content i in T. Then $\psi_T(\alpha) = \prod_i \psi_{\nu_{i+1}/\nu_i}(\alpha)$ where

$\psi_{\lambda/\mu}(\alpha) = \prod_{s \in R_{\lambda/\mu}-C_{\lambda/\mu} } \frac{(\alpha a_\mu(s) + l_\mu(s) +1)}{(\alpha a_\mu(s) + l_\mu(s) + \alpha)} \frac{(\alpha a_\lambda(s) + l_\lambda(s) + \alpha)}{(\alpha a_\lambda(s) + l_\lambda(s) +1)}$

and the product is taken only over all boxes s in $\lambda$ such that s has a box from $\lambda/\mu$ in the same row, but not in the same column.

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

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

References

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External links

• Software for computing the Jack function by Plamen Koev and Alan Edelman.
• MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package)
• SAGE documentation for Jack Symmetric Functions

• Knop & Sahi 1997.
• Macdonald 1995, pp. 379.