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 \kappa, parameter \alpha, and arguments can be recursively defined as follows: where the summation is over all partitions \mu such that the skew partition \kappa/\mu is a horizontal strip, namely where equals if and otherwise. The expressions \kappa' and \mu' refer to the conjugate partitions of \kappa and \mu, respectively. The notation means that the product is taken over all coordinates (i,j) of boxes in the Young diagram of the partition \kappa.

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 \lambda, and with An admissible tableau of shape \lambda is a filling of the Young diagram \lambda with numbers 1,2,…,n such that for any box (i,j) in the tableau, A box is critical for the tableau T if j > 1 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 where a_\lambda and l_\lambda denotes the arm and leg length respectively. Therefore, for is the usual Schur function. Similar to Schur polynomials, P_\lambda 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 \alpha. Thus, a formula for the Jack function P_\lambda is given by where the sum is taken over all tableaux of shape \lambda, and T(s) denotes the entry in box s of T. The weight can be defined in the following fashion: Each tableau T of shape \lambda 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 \lambda such that s has a box from \lambda/\mu in the same row, but not in the same column.

Connection with the Schur polynomial

When \alpha=1 the Jack function is a scalar multiple of the Schur polynomial where is the product of all hook lengths of \kappa.

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 X is a matrix with eigenvalues , then