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
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 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 in the tableau,
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 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