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 $J_{\kappa }^{(\alpha )}(x_1,x_2,\dots ,x_m)$ of an integer partition $\kappa$, parameter $\alpha$, and arguments $x_1,x_2,\dots ,x_m$ 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

$J_{\kappa }^{(\alpha )}(x_1,x_2,\dots ,x_m)=\sum _{\mu }{J}_{\mu }^{(\alpha )}(x_1,x_2,\dots ,x_{m-1})x_m^{|\kappa /\mu |}{\beta }_{\kappa \mu },$

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 $J_{\mu }(x_1,\dots ,x_{n-1})=0$) 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 ${{\kappa }_j}^{\prime }-i+\alpha ({\kappa }_i-j+1)$ if ${{\kappa }_j}^{\prime }={{\mu }_j}^{\prime }$ and ${{\kappa }_j}^{\prime }-i+1+\alpha ({\kappa }_i-j)$ otherwise. The expressions ${\kappa }^{\prime }$ and ${\mu }^{\prime }$ 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,

• $T(i,j)\ne T(i^{\prime },j)$ whenever $i^{\prime }>i.$
• $T(i,j)\ne T(i,j-1)$ whenever $j>1$ and $i^{\prime }<i.$

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:

$⟨f,g⟩=\underset{[0,2\pi {]}^n}{\int }f(e^{i{\theta }_1},\dots ,e^{i{\theta }_n})\overline{g(e^{i{\theta }_1},\dots ,e^{i{\theta }_n})}\prod _{1\le j<k\le n}{|e^{i{\theta }_j}-e^{i{\theta }_k}|}^{\frac{2}{\alpha }}d{\theta }_1\cdots d{\theta }_n$

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

$C_{\kappa }^{(\alpha )}(x_1,\dots ,x_n)=\frac{{\alpha }^{|\kappa |}(|\kappa |)!}{j_{\kappa }}{J}_{\kappa }^{(\alpha )}(x_1,\dots ,x_n),$

where

$j_{\kappa }=\prod _{(i,j)\in \kappa }({{\kappa }_j}^{\prime }-i+\alpha ({\kappa }_i-j+1))({{\kappa }_j}^{\prime }-i+1+\alpha ({\kappa }_i-j)).$

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

P normalization

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

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

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

$\mathrm{\varnothing }={\nu }_1\to {\nu }_2\to \dots \to {\nu }_n=\lambda$

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

$J_{\kappa }^{(1)}(x_1,x_2,\dots ,x_n)=H_{\kappa }{s}_{\kappa }(x_1,x_2,\dots ,x_n),$

where

$H_{\kappa }=\prod _{(i,j)\in \kappa }{h}_{\kappa }(i,j)=\prod _{(i,j)\in \kappa }({\kappa }_i+{{\kappa }_j}^{\prime }-i-j+1)$

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:

$J_{\kappa }^{(\alpha )}(x_1,x_2,\dots ,x_m)=0,\text{ if }{\kappa }_{m+1}>0.$

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 $x_1,x_2,\dots ,x_m$, then

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

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) Archived 2010-06-20 at the Wayback Machine
• SAGE documentation for Jack Symmetric Functions