Jack Polynomial

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 $J_\kappa^(x_1,x_2,\ldots,x_m)$
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 :

: $J_^(x_1)=x_1^k(1+\alpha)\cdots (1+(k-1)\alpha)$

; For ''m''>1:

: $J_\kappa^(x_1,x_2,\ldots,x_m)=\sum_\mu J_\mu^(x_1,x_2,\ldots,x_) x_m^\beta_,$

where the summation is over all partitions $\mu$ such that the skew partition $\kappa/\mu$ is a horizontal strip, namely
:$\kappa_1\ge\mu_1\ge\kappa_2\ge\mu_2\ge\cdots\ge\kappa_\ge\mu_\ge\kappa_n$ ($\mu_n$ must be zero or otherwise $J_\mu(x_1,\ldots,x_)=0$) and
:$\beta_=\frac,$

where $B_^\nu(i,j)$ 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 $J_\mu^$ in ''n'' variables:

:$J_\mu^ = \sum_ d_T(\alpha) \prod_ x_.$

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

:$d_T(\alpha) = \prod_ d_\lambda(\alpha)(s)$

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:

:$\langle f,g\rangle = \int_ f \left (e^,\ldots,e^ \right ) \overline \prod_ \left , e^-e^ \right , ^ d\theta_1\cdots d\theta_n$

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

:$C_\kappa^(x_1,\ldots,x_n) = \frac J_\kappa^(x_1,\ldots,x_n),$

where

:$j_\kappa=\prod_ \left (\kappa_j'-i+\alpha \left (\kappa_i-j+1 \right ) \right ) \left (\kappa_j'-i+1+\alpha \left (\kappa_i-j \right ) \right ).$

For $\alpha=2, C_\kappa^(x_1,\ldots,x_n)$ 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

:$H'_\lambda = \prod_ (\alpha a_\lambda(s) + l_\lambda(s) + 1)$

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

:$P_\lambda = \sum_ \psi_T(\alpha) \prod_ x_$

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 $\nu_/\nu_i$ defines the skew shape with content ''i'' in ''T''. Then

:$\psi_T(\alpha) = \prod_i \psi_(\alpha)$

where

:$\psi_(\alpha) = \prod_ \frac \frac$

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

:$J^_\kappa(x_1,x_2,\ldots,x_n) = H_\kappa s_\kappa(x_1,x_2,\ldots,x_n),$
where
:$H_\kappa=\prod_ h_\kappa(i,j)= \prod_ (\kappa_i+\kappa_j'-i-j+1)$
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:

:$J_\kappa^(x_1,x_2,\ldots,x_m)=0, \mbox\kappa_>0.$

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

:$J_\kappa^(X)=J_\kappa^(x_1,x_2,\ldots,x_m).$

References

*.
*.
*
*
*{{citation
, last = Stanley , first = Richard P. , authorlink = Richard P. Stanley
, doi = 10.1016/0001-8708(89)90015-7
, doi-access=free
, mr = 1014073
, issue = 1
, journal = Advances in Mathematics
, pages = 76–115
, title = Some combinatorial properties of Jack symmetric functions
, volume = 77
, year = 1989.

External links

Software for computing the Jack function
by Plamen Koev and Alan Edelman.


* ttp://www.sagemath.org/doc/reference/sage/combinat/sf/jack.html SAGE documentation for Jack Symmetric Functions Orthogonal polynomials
Special functions
Symmetric functions