Jack function

In mathematics, the Jack function is a homogeneous, symmetric polynomial which generalizes the Schur polynomial.

Definition

The Jack function $J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})$ of integer partition $\kappa$, parameter $\alpha$ and arguments $x_{1},x_{2},\ldots ,x_{n}$ 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 $m>1$:

$J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=\sum _{\mu \leq \kappa }J_{\mu }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m-1})x_{m}^{|\kappa /\mu |}\beta _{\kappa \mu },$ where the summation is over all partitions $\mu$ such that $\mu _{i}\leq \kappa _{i}$ for all $i$ (this is what the expression $\mu \leq \kappa$ means) 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 $\kappa$.

Properties

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

$J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})=0,{\mbox{ if }}\kappa _{n+1}>0.$

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, 2005.

• 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

• Sofware for computing the Jack function by Plamen Koev.