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Wikipedia / 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

of an integer partition can be recursively defined as follows:

  1. For m=1 :

    For m>1:

such that the skew partition is a horizontal strip, namely

  1. ) and

of boxes in the Young diagram.

Combinatorial formula

In 1997, F. Knop and S. Sahi in n variables:

The sum is taken over all admissible and

with

An admissible with numbers 1,2,...,n such that for any box (i,j) in the tableau,

is critical for the tableau T

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

and called the Zonal polynomial.

P normalization

The P, where

denotes the arm and leg length is the usual Schur function.

.

Thus, a formula is given by

denotes the entry in box s of T.

can be defined in the following fashion: Each tableau T can be interpreted as a sequence of partitions

defines the skew shape with content i in T. Then

where

and the product is taken only over all boxes s such that s in the same row, but not in the same column.

Connection with the Schur polynomial

the Jack function is a scalar multiple of the Schur polynomial

where

.

Properties

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

Matrix argument

, then

Bibliography

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References