Comments

The files below contains results related to properties of Jack polynomials and their particular cases (Zonal polynomials, Quaternionic (symplectic) polynomials, Schur polynomials, etc).

Many results were obtained by using the Mathematica package SymPol$Package also available on this web site (there).

The software (code) and the results displayed in the files available on this personal web site can be freely downloaded but they are provided "as they are" without warranty of any kind about their reliability. They should only be considered as private working documents that are made available on line by their author.

Contents

Structure constants of zonal polynomials (examples)

The results given in ZonalStructureConstantsResults are mentioned and used in the article :
" The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices"
by R. Coquereaux and J.-B. Zuber (September 2018), link to arxiv

Mathematica Notebook ZonalStructureConstantsResults.nb
html file ZonalStructureConstantsResults.html
pdf file ZonalStructureConstantsResults.pdf
text file ZonalStructureConstantsResults.txt
rtf file ZonalStructureConstantsResults.rtf

Structure constants of quaternionic (zonal symplectic) polynomials

Structure constants of Jack polynomials

Decomposition of the square of the 3-variables Jack polynomial (P normalization) defined by the partition {2s, s, 0} for s=3, reduction to SU(3) is performed, the file displays a ListPlot3D of the structure constants with a Manipulate using the varying Jack parameter alpha.

Mathematica Notebook: Manipulate ListPlot3D of square of s[1,1] for s = 3. ManipulateAdjointSquare.nb
CDF file (for Wolfram Player): Manipulate ListPlot3D of square of s[1,1] for s = 3. ManipulateAdjointSquare.cdf

Plot the terms that appear in the decomposition of the (zonal) square of [s,s] = {2s,s,0}

Each term is displayed as a point in the Horn curved polygon (use variables P,Q). Mathematica Notebook