Jack function

The Jack function $J\_\backslash kappa^\{(\backslash alpha\; )\}(x\_1,x\_2,\backslash ldots,x\_m)$

of an integer partition $\backslash kappa$, parameter $\backslash alpha$, and arguments $x\_1,x\_2,\backslash ldots,x\_m$ can be recursively defined as

follows:

; For m=1 :

: $J\_\{k\}^\{(\backslash alpha\; )\}(x\_1)=x\_1^k(1+\backslash alpha)\backslash cdots\; (1+(k-1)\backslash alpha)$

; For m>1:

: $J\_\backslash kappa^\{(\backslash alpha\; )\}(x\_1,x\_2,\backslash ldots,x\_m)=\backslash sum\_\backslash mu$

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 $\backslash mu$ such that the skew partition $\backslash kappa/\backslash mu$ 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

($\backslash mu\_n$ must be zero or otherwise $J\_\backslash mu(x\_1,\backslash ldots,x\_\{n-1\})=0$) 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 $B\_\{\backslash kappa\backslash mu\}^\backslash nu(i,j)$ equals $\backslash kappa\_j\text{'}-i+\backslash alpha(\backslash kappa\_i-j+1)$ if $\backslash kappa\_j\text{'}=\backslash mu\_j\text{'}$ and $\backslash kappa\_j\text{'}-i+1+\backslash alpha(\backslash kappa\_i-j)$ otherwise. The expressions $\backslash kappa\text{'}$ and $\backslash mu\text{'}$ refer to the conjugate partitions of $\backslash kappa$ and $\backslash mu$, respectively. The notation $(i,j)\backslash in\backslash kappa$ means that the product is taken over all coordinates $(i,j)$ of boxes in the Young diagram of the partition $\backslash kappa$.

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

:$J\_\backslash mu^\{(\backslash alpha\; )\}\; =\; \backslash sum\_\{T\}\; d\_T(\backslash alpha)\; \backslash prod\_\{s\; \backslash in\; T\}\; x\_\{T(s)\}.$

The sum is taken over all admissible tableaux of shape $\backslash lambda,$ and

:$d\_T(\backslash alpha)\; =\; \backslash prod\_\{s\; \backslash in\; T\; \backslash text\{\; critical\}\}\; d\_\backslash lambda(\backslash alpha)(s)$

with

:$d\_\backslash lambda(\backslash alpha)(s)\; =\; \backslash alpha(a\_\backslash lambda(s)\; +1)\; +\; (l\_\backslash lambda(s)\; +\; 1).$

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

• $T(i,j)\; \backslash neq\; T(i\text{'},j)$ whenever $i\text{'}>i.$
• $T(i,j)\; \backslash neq\; T(i,j-1)$ whenever $j>1$ and

A box $s\; =\; (i,j)\; \backslash in\; \backslash lambda$ 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.

The Jack functions form an orthogonal basis in a space of symmetric polynomials, with inner product:

:

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

:$C\_\backslash kappa^\{(\backslash alpha)\}(x\_1,\backslash ldots,x\_n)\; =\; \backslash frac\{\backslash alpha^$

\kappa

(|\kappa|)!}{j_\kappa} J_\kappa^{(\alpha)}(x_1,\ldots,x_n),

where

:$j\_\backslash kappa=\backslash prod\_\{(i,j)\backslash in\; \backslash kappa\}\; \backslash left\; (\backslash kappa\_j\text{'}-i+\backslash alpha\; \backslash left\; (\backslash kappa\_i-j+1\; \backslash right\; )\; \backslash right\; )\; \backslash left\; (\backslash kappa\_j\text{'}-i+1+\backslash alpha\; \backslash left\; (\backslash kappa\_i-j\; \backslash right\; )\; \backslash right\; ).$

For $\backslash alpha=2,\; C\_\backslash kappa^\{(2)\}(x\_1,\backslash ldots,x\_n)$ is often denoted by $C\_\backslash kappa(x\_1,\backslash ldots,x\_n)$ and called the Zonal polynomial.

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

:$H\text{'}\_\backslash lambda\; =\; \backslash prod\_\{s\backslash in\; \backslash lambda\}\; (\backslash alpha\; a\_\backslash lambda(s)\; +\; l\_\backslash lambda(s)\; +\; 1)$

where $a\_\backslash lambda$ and $l\_\backslash lambda$ denotes the arm and leg length respectively. Therefore, for $\backslash alpha=1,\; P\_\backslash lambda$ is the usual Schur function.

Similar to Schur polynomials, $P\_\backslash lambda$ 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 $\backslash alpha$.

Thus, a formula {{sfn|Macdonald|1995|pp=379}} for the Jack function $P\_\backslash lambda$ is given by

:$P\_\backslash lambda\; =\; \backslash sum\_\{T\}\; \backslash psi\_T(\backslash alpha)\; \backslash prod\_\{s\; \backslash in\; \backslash lambda\}\; x\_\{T(s)\}$

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

The weight $\backslash psi\_T(\backslash alpha)$ can be defined in the following fashion: Each tableau T of shape $\backslash lambda$ can be interpreted as a sequence of partitions

:$\backslash emptyset\; =\; \backslash nu\_1\; \backslash to\; \backslash nu\_2\; \backslash to\; \backslash dots\; \backslash to\; \backslash nu\_n\; =\; \backslash lambda$

where $\backslash nu\_\{i+1\}/\backslash nu\_i$ defines the skew shape with content i in T. Then

:$\backslash psi\_T(\backslash alpha)\; =\; \backslash prod\_i\; \backslash psi\_\{\backslash nu\_\{i+1\}/\backslash nu\_i\}(\backslash alpha)$

where

:$\backslash psi\_\{\backslash lambda/\backslash mu\}(\backslash alpha)\; =\; \backslash prod\_\{s\; \backslash in\; R\_\{\backslash lambda/\backslash mu\}-C\_\{\backslash lambda/\backslash mu\}\; \}\; \backslash frac\{(\backslash alpha\; a\_\backslash mu(s)\; +\; l\_\backslash mu(s)\; +1)\}\{(\backslash alpha\; a\_\backslash mu(s)\; +\; l\_\backslash mu(s)\; +\; \backslash alpha)\}\; \backslash frac\{(\backslash alpha\; a\_\backslash lambda(s)\; +\; l\_\backslash lambda(s)\; +\; \backslash alpha)\}\{(\backslash alpha\; a\_\backslash lambda(s)\; +\; l\_\backslash lambda(s)\; +1)\}$

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

When $\backslash alpha=1$ 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 $\backslash kappa$.

If the partition has more parts than the number of variables, then the Jack function is 0:

:$J\_\backslash kappa^\{(\backslash alpha\; )\}(x\_1,x\_2,\backslash ldots,x\_m)=0,\; \backslash mbox\{\; if\; \}\backslash kappa\_\{m+1\}>0.$

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

$x\_1,x\_2,\backslash ldots,x\_m$, then

:$$

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

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• [http://www-math.mit.edu/~plamen/software Software for computing the Jack function] by Plamen Koev and Alan Edelman.
• [http://www.math.washington.edu/~dumitriu/mopspage.html MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package)] {{Webarchive|url=https://web.archive.org/web/20100620202845/http://www.math.washington.edu/~dumitriu/mopspage.html |date=2010-06-20 }}
• [http://www.sagemath.org/doc/reference/sage/combinat/sf/jack.html SAGE documentation for Jack Symmetric Functions]

Category:Orthogonal polynomials

Category:Special functions

Category:Symmetric functions