Jack function

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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[edit]

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 .

Combinatorial formula[edit]

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 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[edit]

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[edit]

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 [2] 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[edit]

When the Jack function is a scalar multiple of the Schur polynomial

where

is the product of all hook lengths of .

Properties[edit]

If the partition has more parts than the number of variables, then the Jack function is 0:

Matrix argument[edit]

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

References[edit]

  • Demmel, James; Koev, Plamen (2006), "Accurate and efficient evaluation of Schur and Jack functions", Mathematics of Computation, 75 (253): 223–239, CiteSeerX 10.1.1.134.5248, 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, arXiv:q-alg/9610016, Bibcode:1997InMat.128....9K, doi:10.1007/s002220050134
  • Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), New York: Oxford University Press, ISBN 978-0-19-853489-1, 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.

External links[edit]

  • ^ Knop & Sahi 1997.
  • ^ Macdonald 1995, pp. 379.