Jack function

In mathematics, the Jack function is a generalization of the Jack polynomial, introduced by Henry Jack. The Jack polynomial 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 an integer partition $$\kappa$$, parameter $$\alpha$$, and arguments $$x_1,x_2,\ldots,x_m$$ 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,
 * $$T(i,j) \neq T(i',j)$$ whenever $$i'>i.$$
 * $$T(i,j) \neq T(i,j-1)$$ whenever $$j>1$$ and $$i' 1$$ 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:


 * $$\langle f,g\rangle = \int_{[0,2\pi]^n} f \left (e^{i\theta_1},\ldots,e^{i\theta_n} \right ) \overline{g \left (e^{i\theta_1},\ldots,e^{i\theta_n} \right )} \prod_{1\le j<k\le n} \left |e^{i\theta_j}-e^{i\theta_k} \right |^{\frac{2}{\alpha}} d\theta_1\cdots d\theta_n$$

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


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

where


 * $$j_\kappa=\prod_{(i,j)\in \kappa} \left (\kappa_j'-i+\alpha \left (\kappa_i-j+1 \right ) \right ) \left (\kappa_j'-i+1+\alpha \left (\kappa_i-j \right ) \right ).$$

For $$\alpha=2, C_\kappa^{(2)}(x_1,\ldots,x_n)$$ is often denoted by $$C_\kappa(x_1,\ldots,x_n)$$ and called the Zonal polynomial.

P normalization
The P normalization is given by the identity $$J_\lambda = H'_\lambda P_\lambda$$, where


 * $$H'_\lambda = \prod_{s\in \lambda} (\alpha a_\lambda(s) + l_\lambda(s) + 1)$$

where $$a_\lambda$$ 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). $$