Jack polynomials and free cumulants

  This page gives new data for Jack polynomials. Our results have been published there.
 
  Being given some parameter &alpha and an arbitrary partition &lambda, we consider the Jack polynomial associated to &lambda, and its development in terms of the power sum symmetric functions, i.e. we write

  For &mu with no part 1 and weight k, we give the explicit expression of the coefficients
 
in terms of the free cumulants of the anisotropic diagram of &lambda.
  These coefficients are known to be polynomials in the free cumulants. We list them

• for any partition &mu, with weight(&mu) - length(&mu) < 9,
• when &mu is a hook (r,1,...,1), for r from 2 to 20,
• when &mu=(r,s) has length 2, for r+s from 4 to 18.

  Our data support the following conjectures :

• These coefficients are polynomials in &alpha and &beta = 1- &alpha, with integer coefficients.
• When &mu is a hook, their integer coefficients are nonnegative.
• When &mu is not a hook, their integer coefficients may be negative but an appropriately modified polynomial has still nonnegative coefficients.

  These conjectures extend the Kerov-Biane ex-conjecture for characters of the symmetric group, recently proved by Feray.

  Tables giving &theta (&lambda,&mu)

• for weight(&mu) - length(&mu) from 1 to 8
• for hooks with r from 2 to 20.

  When &mu is not a hook, tables giving the modified &theta (&lambda,&mu) for partitions &mu=(r,s) with r+s from 4 to 18
(0.5 Mo).

  Tables for bigger values may be given upon request.

  For &alpha = 1 ( i.e. a = 1 and b = 0 ), these tables give the Kerov-Biane polynomials , expressing the normalized characters of the symmetric group.

  Our results are in Maple format. They should be read as follows.

• The parameter &alpha is denoted by the letter a, and the letter b stands for &beta = 1-&alpha.
• The partition &lambda is kept arbitrary.
  The notation R_k stands for the k-th free cumulant of the anisotropic diagram of &lambda.
  
• &mu denotes a partition without any part 1.
  
• Each table gives first the partition &mu, then
  

Example :
[2]
a*b*R2+a^2*R3
[3]
2*a*b^2*R2+a^2*R2+3*a^2*b*R3+a^3*R4
means

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