# 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.

## Definition

The Jack function of an integer partition , parameter , and indefinitely many arguments can be recursively defined as follows:

- For
*m*=1

- For
*m*>1

where the summation is over all partitions such that the **skew partition** is a **horizontal strip**, namely

- ( must be zero or otherwise ) and

where equals if and otherwise. The expressions and refer to the conjugate partitions of and , respectively. The notation means that the product is taken over all coordinates of boxes in the Young diagram of the partition .

### Combinatorial formula

In 1997, F. Knop and S. Sahi [1] gave a purely combinatorial formula for the Jack polynomials in *n* variables:

The sum is taken over all *admissible* tableaux of shape and

with

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

- whenever
- whenever and

A box is *critical* for the tableau *T* if and

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:

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

where

For is often denoted by and called the Zonal polynomial.

## P normalization

The *P* normalization is given by the identity , where

and and denotes the arm and leg length respectively. Therefore, for is the usual Schur function.

Similar to Schur polynomials, 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 .

Thus, a formula [2] for the Jack function is given by

where the sum is taken over all tableaux of shape , and denotes the entry in box *s* of *T*.

The weight can be defined in the following fashion: Each tableau *T* of shape can be interpreted as a sequence of partitions

where defines the skew shape with content *i* in *T*. Then

where

and the product is taken only over all boxes *s* in such that *s* has a box from in the same row, but *not* in the same column.

## Connection with the Schur polynomial

When the Jack function is a scalar multiple of the Schur polynomial

where

is the product of all hook lengths of .

## Properties

If the partition has more parts than the number of variables, then the Jack function is 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 is a matrix with eigenvalues , then

## References

- 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 - 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.

## External links

- Software for computing the Jack function by Plamen Koev and Alan Edelman.
- MOPS: Multivariate Orthogonal Polynomials (symbolically) (Maple Package)
- SAGE documentation for Jack Symmetric Functions

- Knop & Sahi 1997.
- Macdonald 1995, pp. 379.