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, i.e.
  
  
• &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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