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.
of an integer partition
can be recursively defined as follows: where the summation is over all partitions
κ μ
refer to the conjugate partitions of
means that the product is taken over all coordinates
of boxes in the Young diagram of the partition
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
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.
and called the Zonal polynomial.
denotes the arm and leg length respectively.
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
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
the Jack function is a scalar multiple of the Schur polynomial where 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.
is a matrix with eigenvalues