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 indefinitely many 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

$\kappa _{1}\geq \mu _{1}\geq \kappa _{2}\geq \mu _{2}\geq \cdots \geq \kappa _{n-1}\geq \mu _{n-1}\geq \kappa _{n}$ ($\mu _{n}$ 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' 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\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\leq 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

$C_{\kappa }^{(\alpha )}(x_{1},\ldots ,x_{n})={\frac {\alpha ^{|\kappa |}(|\kappa |)!}{j_{\kappa }}}J_{\kappa }^{(\alpha )}(x_{1},\ldots ,x_{n}),$ 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

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

$J_{\kappa }^{(1)}(x_{1},x_{2},\ldots ,x_{n})=H_{\kappa }s_{\kappa }(x_{1},x_{2},\ldots ,x_{n}),$ 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

$J_{\kappa }^{(\alpha )}(X)=J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m}).$ 