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

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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

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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

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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

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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 [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

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

 

where

 

is the product of all hook lengths of  .

Properties

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

 

Matrix argument

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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

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  • 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, S2CID 7188322
  • 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.
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