Jack polynomials and alpha-contents

  This page gives new data for Jack polynomials.

  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

  We give the explicit expression of the coefficients theta (lambda,mu) appearing in this development, in terms of the alpha-contents of lambda.

  We list these coefficients for any partition mu, such that weight(mu) - length(mu) < 11.
 
  Thanks are due to Alain Lascoux for implementing our method on computer, using ACE.

Tables giving theta (lambda,mu) for weight(mu) - length(mu)

• from 1 to 7
• from 8 to 9
• equal to 10

  Tables for bigger values may be given upon request.

  For alpha = 1 ( i.e. a = 1 and b = 0 ), these tables give the central 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 (alpha - 1).
• The partition lambda is kept arbitrary.
  The letter W stands for the weight of lambda, and p_k (k > 0) stands for the k-th power sum of the alpha-contents of lambda
  
  
• mu denotes a partition without any part 1.
  
• Each table gives first the partition mu, then
  

Example :
[2]
a*p1
[3]
-a*b*p1+a^2*p2+a*(1/2*W-1/2*W^2)
means

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