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A positivity conjecture for Jack polynomials
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Page of Michel Lassalle
List of publications
Some preprints
Tables for characters of the symmetric group
Symmetric functions and Jucys-Murphy elements
Tables for Jack polynomials :
Jack polynomials and alpha-contents
Jack polynomials and free cumulants
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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
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and &mu has no part 1 and weight k,
the coefficients
are polynomials in the variables
with nonnegative integer coefficients.
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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.
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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.
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Last modified : March 8, 2007
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