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 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 .
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,
- T(i,j) ≠ T( i',j) whenever i' > i.
- T(i,j) ≠ T( i',j-1) whenever j>1 and i' < i.
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.
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 denoted often as just is known as 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 .
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
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
- Demmel, James; Koev, Plamen (2006), "Accurate and efficient evaluation of Schur and Jack functions", Mathematics of Computation, 75 (253): 223–239, doi:10.1090/S0025-5718-05-01780-1, MR 2176397.
- Jack, Henry (1970–1971), "A class of symmetric polynomials with a parameter", Proceedings of the Royal Society of Edinburgh, Section A. Mathematics, 69: 1–18, MR 0289462.
- Knop, Friedrich; Sahi, Siddhartha (19 March 1997), "A recursion and a combinatorial formula for Jack polynomials", Inventiones Mathematicae, 128 (1): 9–22, doi:10.1007/s002220050134
- Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), New York: Oxford University Press, ISBN 0-19-853489-2, MR 1354144
- Stanley, Richard P. (1989), "Some combinatorial properties of Jack symmetric functions", Advances in Mathematics, 77 (1): 76–115, doi:10.1016/0001-8708(89)90015-7, MR 1014073.