

A positivity conjecture for Jack polynomials

Page of Michel Lassalle
List of publications
Some preprints
Tables for characters of the symmetric group
Symmetric functions and JucysMurphy elements
Tables for Jack polynomials :
Jack polynomials and alphacontents
Jack polynomials and free cumulants


This page gives data supporting the following conjecture for Jack polynomials, which generalizes Stanley's exconjecture 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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