A positivity conjecture for Jack polynomials

  Page of Michel Lassalle
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characters of the
symmetric group

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  This page gives data supporting the following conjecture for Jack polynomials, which generalizes Stanley's ex-conjecture for normalized characters of the symmetric group.
  Being given some parameter &alpha and an arbitrary partition &lambda, we consider the Jack polynomial associated with &lambda, and its development in terms of the power sum symmetric functions, i.e. we write

  In a recent paper we conjecture that, when &lambda is formed of m rectangular blocks, i.e. has the shape
and &mu has no part 1 and weight k, the coefficients
  are polynomials in the variables
  with nonnegative integer coefficients.

  Here we list these coefficients for m=3, and for any partition &mu without any part 1, such that weight(&mu) - length(&mu) < 6.
  Table giving the coefficients for weight(&mu) - length(&mu) from 1 to 5
(1.7 Mo).
  Actually we have computed these quantities for weight(&mu) - length(&mu) < 9. They are available upon request.

Our results are in Maple format. They should be read as follows.
  • The letter b stands for the parameter (&alpha - 1).
  • The partition &lambda is formed of three rectangular blocks (P1,Q1), (P2,Q2) and (P3,Q3).
  • &mu denotes a partition without any part 1.
  • The table gives first the partition &mu, then

    as a polynomial in b, P1, -Q1, P2, -Q2, P3, -Q3.

Last modified : March 8, 2007

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