Jack function

In mathematics, the Jack function, introduced by Henry Jack, is a homogenous, 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 $\alpha$ and arguments $x_{1},x_{2},\ldots ,$ 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

$J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=\sum _{\mu }J_{\mu }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m-1})x_{m}^{|\kappa /\mu |}\beta _{\kappa \mu },$

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 $\kappa$.

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

$C_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{n})={\frac {\alpha ^{|\kappa |}(|\kappa |)!}{j_{\kappa }}}J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{n}),$

where

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

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

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

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

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

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

• Software for computing the Jack function by Plamen Koev.