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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