Jack function

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.

The Jack function ${\displaystyle J_{\kappa }^{(\alpha )}(x_{1},x_{2},\ldots ,x_{m})}$ of an integer partition ${\displaystyle \kappa }$, parameter ${\displaystyle \alpha }$, and arguments ${\displaystyle x_{1},x_{2},\ldots ,x_{m}}$ can be recursively defined as follows:

For m=1

${\displaystyle J_{k}^{(\alpha )}(x_{1})=x_{1}^{k}(1+\alpha )\cdots (1+(k-1)\alpha )}$

For m>1

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

where the summation is over all partitions ${\displaystyle \mu }$ such that the skew partition ${\displaystyle \kappa /\mu }$ is a horizontal strip, namely

${\displaystyle \kappa _{1}\geq \mu _{1}\geq \kappa _{2}\geq \mu _{2}\geq \cdots \geq \kappa _{n-1}\geq \mu _{n-1}\geq \kappa _{n}}$ (${\displaystyle \mu _{n}}$ must be zero or otherwise ${\displaystyle J_{\mu }(x_{1},\ldots ,x_{n-1})=0}$) and

${\displaystyle \beta _{\kappa \mu }={\frac {\prod _{(i,j)\in \kappa }B_{\kappa \mu }^{\kappa }(i,j)}{\prod _{(i,j)\in \mu }B_{\kappa \mu }^{\mu }(i,j)}},}$

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

In 1997, F. Knop and S. Sahi ^[1] gave a purely combinatorial formula for the Jack polynomials ${\displaystyle J_{\mu }^{(\alpha )}}$ in n variables:

${\displaystyle J_{\mu }^{(\alpha )}=\sum _{T}d_{T}(\alpha )\prod _{s\in T}x_{T(s)}.}$

The sum is taken over all admissible tableaux of shape ${\displaystyle \lambda ,}$ and

${\displaystyle d_{T}(\alpha )=\prod _{s\in T{\text{ critical}}}d_{\lambda }(\alpha )(s)}$

with

${\displaystyle d_{\lambda }(\alpha )(s)=\alpha (a_{\lambda }(s)+1)+(l_{\lambda }(s)+1).}$

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

• ${\displaystyle T(i,j)\neq T(i',j)}$ whenever ${\displaystyle i'>i.}$
• ${\displaystyle T(i,j)\neq T(i,j-1)}$ whenever ${\displaystyle j>1}$ and ${\displaystyle i'<i.}$

A box ${\displaystyle s=(i,j)\in \lambda }$ is critical for the tableau T if ${\displaystyle j>1}$ and ${\displaystyle T(i,j)=T(i,j-1).}$

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:

${\displaystyle \langle f,g\rangle =\int _{[0,2\pi ]^{n}}f\left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right){\overline {g\left(e^{i\theta _{1}},\ldots ,e^{i\theta _{n}}\right)}}\prod _{1\leq j<k\leq n}\left|e^{i\theta _{j}}-e^{i\theta _{k}}\right|^{\frac {2}{\alpha }}d\theta _{1}\cdots d\theta _{n}}$

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

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

where

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

For ${\displaystyle \alpha =2,C_{\kappa }^{(2)}(x_{1},\ldots ,x_{n})}$ is often denoted by ${\displaystyle C_{\kappa }(x_{1},\ldots ,x_{n})}$ and called the Zonal polynomial.

The P normalization is given by the identity ${\displaystyle J_{\lambda }=H'_{\lambda }P_{\lambda }}$, where

${\displaystyle H'_{\lambda }=\prod _{s\in \lambda }(\alpha a_{\lambda }(s)+l_{\lambda }(s)+1)}$

where ${\displaystyle a_{\lambda }}$ and ${\displaystyle l_{\lambda }}$ denotes the arm and leg length respectively. Therefore, for ${\displaystyle \alpha =1,P_{\lambda }}$ is the usual Schur function.

Similar to Schur polynomials, ${\displaystyle P_{\lambda }}$ 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 ${\displaystyle \alpha }$.

Thus, a formula ^[2] for the Jack function ${\displaystyle P_{\lambda }}$ is given by

${\displaystyle P_{\lambda }=\sum _{T}\psi _{T}(\alpha )\prod _{s\in \lambda }x_{T(s)}}$

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

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

${\displaystyle \emptyset =\nu _{1}\to \nu _{2}\to \dots \to \nu _{n}=\lambda }$

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

${\displaystyle \psi _{T}(\alpha )=\prod _{i}\psi _{\nu _{i+1}/\nu _{i}}(\alpha )}$

where

${\displaystyle \psi _{\lambda /\mu }(\alpha )=\prod _{s\in R_{\lambda /\mu }-C_{\lambda /\mu }}{\frac {(\alpha a_{\mu }(s)+l_{\mu }(s)+1)}{(\alpha a_{\mu }(s)+l_{\mu }(s)+\alpha )}}{\frac {(\alpha a_{\lambda }(s)+l_{\lambda }(s)+\alpha )}{(\alpha a_{\lambda }(s)+l_{\lambda }(s)+1)}}}$

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

When ${\displaystyle \alpha =1}$ the Jack function is a scalar multiple of the Schur polynomial

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

where

${\displaystyle H_{\kappa }=\prod _{(i,j)\in \kappa }h_{\kappa }(i,j)=\prod _{(i,j)\in \kappa }(\kappa _{i}+\kappa _{j}'-i-j+1)}$

is the product of all hook lengths of ${\displaystyle \kappa }$.

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

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

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 ${\displaystyle X}$ is a matrix with eigenvalues ${\displaystyle x_{1},x_{2},\ldots ,x_{m}}$, then

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

• Demmel, James; Koev, Plamen (2006), "Accurate and efficient evaluation of Schur and Jack functions", Mathematics of Computation, 75 (253): 223–239, CiteSeerX 10.1.1.134.5248, doi:10.1090/S0025-5718-05-01780-1, MR 2176397.
• Jack, Henry (1970–1971), "A class of symmetric polynomials with a parameter", Proceedings of the Royal Society of Edinburgh, Section A. Mathematics, 69: 1–18, MR 0289462.
• Knop, Friedrich; Sahi, Siddhartha (19 March 1997), "A recursion and a combinatorial formula for Jack polynomials", Inventiones Mathematicae, 128 (1): 9–22, arXiv:q-alg/9610016, Bibcode:1997InMat.128....9K, doi:10.1007/s002220050134, S2CID 7188322
• Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), New York: Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144
• Stanley, Richard P. (1989), "Some combinatorial properties of Jack symmetric functions", Advances in Mathematics, 77 (1): 76–115, doi:10.1016/0001-8708(89)90015-7, MR 1014073.

• Software for computing the Jack function by Plamen Koev and Alan Edelman.
• MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package) Archived 2010-06-20 at the Wayback Machine
• SAGE documentation for Jack Symmetric Functions