Jack function

In mathematics, the Jack function, introduced by Henry Jack, is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials,and is in turn generalized by the Macdonald polynomials.

Definition

The Jack function $J\_kappa^\{(alpha\; )\}(x\_1,x\_2,ldots,x\_m)$ of integer partition $kappa$, parameter $alpha$ andarguments $x\_1,x\_2,ldots,$ 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 $m>1$:

::

: where the summation is over all partitions $mu$ such that the skew partition $kappa/mu$ is a horizontal strip, namely:$kappa\_1gemu\_1gekappa\_2gemu\_2gecdotsgekappa\_\{n-1\}gemu\_\{n-1\}gekappa\_n$ ($mu\_n$ must be zero or otherwise $J\_mu(x\_1,ldots,x\_\{n-1\})=0$) and ::

: where $B\_\{kappamu\}^\; u(i,j)$ equals $kappa\_j\text{'}-i+alpha(kappa\_i-j+1)$ if $kappa\_j\text{'}=mu\_j\text{'}$ and $kappa\_j\text{'}-i+1+alpha(kappa\_i-j)$ otherwise. The expressions $kappa\text{'}$ and $mu\text{'}$ refer to the conjugate partitions of $kappa$ and $mu$, respectively. The notation $(i,j)inkappa$ means that the product is taken over all coordinates $(i,j)$ of boxes in the Young diagram of the partition $kappa$.

C normalization

The Jack functions form an orthogonal basis in a space of symmetric polynomials. 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:$C\_kappa^\{(alpha)\}(x\_1,x\_2,ldots,x\_n)=frac\{alpha^(|kappa|)!\}\{j\_kappa\}J\_kappa^\{(alpha)\}(x\_1,x\_2,ldots,x\_n),$where:$j\_kappa=prod\_\{(i,j)in\; kappa\}(kappa\_j\text{'}-i+alpha(kappa\_i-j+1))(kappa\_j\text{'}-i+1+alpha(kappa\_i-j)).$

For $alpha=2,;\; C\_kappa^\{(2)\}(x\_1,x\_2,ldots,x\_n)$ denoted often as just$C\_kappa(x\_1,x\_2,ldots,x\_n)$ is known as the Zonal polynomial.

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:$H\_kappa=prod\_\{(i,j)inkappa\}\; h\_kappa(i,j)=prod\_\{(i,j)inkappa\}\; (kappa\_i+kappa\_j\text{'}-i-j+1)$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

:$J\_kappa^\{(alpha\; )\}(X)=J\_kappa^\{(alpha\; )\}(x\_1,x\_2,ldots,x\_m).$

References

* James Demmel and Plamen Koev, "Accurate and efficient evaluation of Schur and Jack functions", "Math. Comp.", 75, no. 253, 223–239, 2006 (article electonically published August 31,2005)
* H. Jack, "A class of symmetric polynomials with a parameter", "Proc. Roy. Soc. Edinburgh Sect. A", 69, 1-18, 1970/1971.
*I. G. Macdonald, "Symmetric functions and Hall polynomials", Second ed., Oxford University Press, New York, 1995.
*Richard Stanley, "Some combinatorial properties of Jack symmetric functions", "Adv. Math.", 77, no. 1, 76–115, 1989.

External links

* [http://www-math.mit.edu/~plamen/software Software for computing the Jack function] by Plamen Koev.