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 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 . 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 (i,j) in the tableau, whenever 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. 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 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