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 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 .
The sum is taken over all admissible tableaux of shape and
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 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.
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
For is often denoted by and called the Zonal polynomial.
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 .
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
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
is the product of all hook lengths of .
If the partition has more parts than the number of variables, then the Jack function is 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 is a matrix with eigenvalues , then
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