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{{Short description|Generalization of the Jack polynomial}}
In [[mathematics]], the '''Jack function''' is a generalization of the '''Jack polynomial''', introduced by [[Henry Jack]]. The Jack polynomial is a [[homogeneous polynomial|homogeneous]], [[symmetric polynomial|symmetric]] [[polynomial]] which generalizes the [[Schur polynomial|Schur]] and [[Zonal polynomial|zonal]] polynomials, and is in turn generalized by the [[Heckman–Opdam polynomials]] and [[Macdonald polynomial]]s.

==Definition==
The Jack function <math>J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m)</math>
of an [[integer partition]] <math>\kappa</math>, parameter <math>\alpha</math>, and arguments <math>x_1,x_2,\ldots,x_m</math> can be recursively defined as
follows:

; For ''m''=1 :

: <math>J_{k}^{(\alpha )}(x_1)=x_1^k(1+\alpha)\cdots (1+(k-1)\alpha)</math>

; For ''m''>1:

: <math>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}, </math>

where the summation is over all partitions <math>\mu</math> such that the '''skew partition''' <math>\kappa/\mu</math> is a '''horizontal strip''', namely
:<math>
\kappa_1\ge\mu_1\ge\kappa_2\ge\mu_2\ge\cdots\ge\kappa_{n-1}\ge\mu_{n-1}\ge\kappa_n
</math> (<math>\mu_n</math> must be zero or otherwise <math>J_\mu(x_1,\ldots,x_{n-1})=0</math>) and
:<math>
\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)
},
</math>

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

===Combinatorial formula===

In 1997, F. Knop and S. Sahi {{sfn|Knop|Sahi|1997}} gave a purely combinatorial formula for the Jack polynomials <math>J_\mu^{(\alpha )}</math> in ''n'' variables:

:<math>J_\mu^{(\alpha )} = \sum_{T} d_T(\alpha) \prod_{s \in T} x_{T(s)}.</math>

The sum is taken over all ''admissible'' tableaux of shape <math>\lambda,</math> and

:<math>d_T(\alpha) = \prod_{s \in T \text{ critical}} d_\lambda(\alpha)(s)</math>

with

:<math>d_\lambda(\alpha)(s) = \alpha(a_\lambda(s) +1) + (l_\lambda(s) + 1).</math>

An ''admissible'' tableau of shape <math>\lambda</math> is a filling of the Young diagram <math>\lambda</math> with numbers 1,2,…,''n'' such that for any box (''i'',''j'') in the tableau,
* <math>T(i,j) \neq T(i',j)</math> whenever <math>i'>i.</math>
* <math>T(i,j) \neq T(i,j-1)</math> whenever <math>j>1</math> and <math>i'<i.</math>

A box <math>s = (i,j) \in \lambda</math> is ''critical'' for the tableau ''T'' if <math>j > 1</math> and <math>T(i,j)=T(i,j-1).</math>

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:

:<math>\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\le j<k\le n} \left |e^{i\theta_j}-e^{i\theta_k} \right |^{\frac{2}{\alpha}} d\theta_1\cdots d\theta_n</math>

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

:<math>C_\kappa^{(\alpha)}(x_1,\ldots,x_n) = \frac{\alpha^{|\kappa|}(|\kappa|)!}{j_\kappa} J_\kappa^{(\alpha)}(x_1,\ldots,x_n),</math>

where

:<math>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 ).</math>

For <math>\alpha=2, C_\kappa^{(2)}(x_1,\ldots,x_n)</math> is often denoted by <math>C_\kappa(x_1,\ldots,x_n)</math> and called the [[Zonal polynomial]].

==P normalization==

The ''P'' normalization is given by the identity <math>J_\lambda = H'_\lambda P_\lambda</math>, where

:<math>H'_\lambda = \prod_{s\in \lambda} (\alpha a_\lambda(s) + l_\lambda(s) + 1)</math>

where <math>a_\lambda</math> and <math>l_\lambda</math> denotes the [[Young tableau#Arm and leg length|arm and leg length]] respectively. Therefore, for <math>\alpha=1, P_\lambda</math> is the usual Schur function.

Similar to Schur polynomials, <math>P_\lambda</math> 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 <math>\alpha</math>.

Thus, a formula {{sfn|Macdonald|1995|pp=379}} for the Jack function <math>P_\lambda </math> is given by

:<math> P_\lambda = \sum_{T} \psi_T(\alpha) \prod_{s \in \lambda} x_{T(s)}</math>

where the sum is taken over all tableaux of shape <math>\lambda</math>, and <math>T(s)</math> denotes the entry in box ''s'' of ''T''.

The weight <math> \psi_T(\alpha) </math> can be defined in the following fashion: Each tableau ''T'' of shape <math>\lambda</math> can be interpreted as a sequence of partitions

:<math> \emptyset = \nu_1 \to \nu_2 \to \dots \to \nu_n = \lambda</math>

where <math>\nu_{i+1}/\nu_i</math> defines the skew shape with content ''i'' in ''T''. Then

:<math> \psi_T(\alpha) = \prod_i \psi_{\nu_{i+1}/\nu_i}(\alpha)</math>

where

:<math>\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)}
</math>

and the product is taken only over all boxes ''s'' in <math>\lambda</math> such that ''s'' has a box from <math>\lambda/\mu</math> in the same row, but ''not'' in the same column.

==Connection with the Schur polynomial==

When <math>\alpha=1</math> the Jack function is a scalar multiple of the [[Schur polynomial]]

:<math>
J^{(1)}_\kappa(x_1,x_2,\ldots,x_n) = H_\kappa s_\kappa(x_1,x_2,\ldots,x_n),
</math>
where
:<math>
H_\kappa=\prod_{(i,j)\in\kappa} h_\kappa(i,j)=
\prod_{(i,j)\in\kappa} (\kappa_i+\kappa_j'-i-j+1)
</math>
is the product of all hook lengths of <math>\kappa</math>.

==Properties==

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

:<math>J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m)=0, \mbox{ if }\kappa_{m+1}>0.</math>

==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 <math>X</math> is a matrix with eigenvalues
<math>x_1,x_2,\ldots,x_m</math>, then

:<math>
J_\kappa^{(\alpha )}(X)=J_\kappa^{(\alpha )}(x_1,x_2,\ldots,x_m).
</math>

==References==

*{{citation
| last1 = Demmel | first1 = James | author1-link = James Demmel
| last2 = Koev | first2 = Plamen
| doi = 10.1090/S0025-5718-05-01780-1
| mr = 2176397
| issue = 253
| journal = [[Mathematics of Computation]]
| pages = 223–239
| title = Accurate and efficient evaluation of Schur and Jack functions
| volume = 75
| year = 2006| citeseerx = 10.1.1.134.5248 }}.
*{{citation
| last = Jack | first = Henry | authorlink = Henry Jack
| mr = 0289462
| journal = Proceedings of the Royal Society of Edinburgh | series = Section A. Mathematics
| pages = 1–18
| title = A class of symmetric polynomials with a parameter
| volume = 69
| year = 1970–1971}}.
*{{citation
|last1=Knop|first1=Friedrich|last2=Sahi|first2=Siddhartha
|title=A recursion and a combinatorial formula for Jack polynomials
|journal=Inventiones Mathematicae
|date=19 March 1997

|volume=128
|issue=1
|pages=9–22
|doi=10.1007/s002220050134|arxiv=q-alg/9610016|bibcode=1997InMat.128....9K|s2cid=7188322 }}
*{{citation
| last = Macdonald | first = I. G. | authorlink = Ian G. Macdonald
| edition = 2nd
| mr = 1354144
| isbn = 978-0-19-853489-1
| location = New York
| publisher = Oxford University Press
| series = Oxford Mathematical Monographs
| title = Symmetric functions and Hall polynomials
| year = 1995}}
*{{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==
* [http://www-math.mit.edu/~plamen/software Software for computing the Jack function] by Plamen Koev and Alan Edelman.
* [http://www.math.washington.edu/~dumitriu/mopspage.html MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package)] {{Webarchive|url=https://web.archive.org/web/20100620202845/http://www.math.washington.edu/~dumitriu/mopspage.html |date=2010-06-20 }}
* [http://www.sagemath.org/doc/reference/sage/combinat/sf/jack.html SAGE documentation for Jack Symmetric Functions]
[[Category:Orthogonal polynomials]]
[[Category:Special functions]]
[[Category:Symmetric functions]]

Jack function - Revision history

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