Page of Michel Lassalle
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characters of the
Jack polynomials :
A conjecture for
This page gives new data for Jack polynomials. Our results have been published there.
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
For &mu with no part 1 and weight k, we give the explicit expression of the coefficients
in terms of the free cumulants of the anisotropic diagram of &lambda.
are known to be polynomials in the free cumulants. We list them
Our data support the following conjectures :
- for any partition &mu, with weight(&mu) - length(&mu) < 9,
- when &mu is a hook (r,1,...,1), for r from 2 to 20,
- when &mu=(r,s) has length 2, for r+s from 4 to 18.
These conjectures extend the Kerov-Biane ex-conjecture for characters of the symmetric group, recently proved by Feray.
- These coefficients are polynomials in &alpha and &beta = 1- &alpha, with integer coefficients.
- When &mu is a hook, their integer coefficients are nonnegative.
- When &mu is not a hook, their integer coefficients may be negative but an appropriately modified polynomial has still nonnegative coefficients.