- Jack function
In
mathematics , the Jack function, introduced by Henry Jack, is a homogeneous, symmetric polynomial which generalizes the Schur and zonal polynomials,and is in turn generalized by theMacdonald polynomial s.Definition
The Jack function of
integer partition , parameter andarguments can be recursively defined as follows:* For :
::
* For :
::
: 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 .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:where:
For denoted often as just is known as the
Zonal polynomial .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
* 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
* [http://www-math.mit.edu/~plamen/software Software for computing the Jack function] by Plamen Koev.
Wikimedia Foundation. 2010.