The results given below 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)

The results given below can be obtained by using the Mathematica package SymPol$Package (the file itself is called SymPol$Package.wl in the same directory).
See also the companion file SymPol$Examples.nb (usage and examples).
See also the file HistogramsUsingSymPol.nb that contains some extra code related to the specific example studied below in the first section (square of {2,1,0} and of its multiples);  in particular this file contains commands for visualization.

The present file only contains results. Most cells are non editable and non evaluatable.

Structure constants of zonal polynomials and SU(n) reduction

Structure constants of zonal polynomials : the square of {2*s, 1*s, 0} -- extended partition -- for s=2,..,6

Consider the following 3-variables zonal polynomial ZP[{2,1,0}], using the P normalization (hence ZP) :

ZonalStructureConstantsResults_1.png

ZonalStructureConstantsResults_2.png

We now consider its square, and decompose on zonal polynomials (same normalization). We get :

ZonalStructureConstantsResults_3.png

Arguments, above and below (in this subsection), denote extended partitions.

More generally we consider the square of the 3-variables zonal polynomials ZP[{2s,s,0}] and obtain :

ZonalStructureConstantsResults_4.png

ZonalStructureConstantsResults_5.png

ZonalStructureConstantsResults_6.png

ZonalStructureConstantsResults_7.png

ZonalStructureConstantsResults_8.png

Structure constants of zonal polynomials reduced to SU(3) : the square of [s,s] -- Dynkin basis -- for s=2,..,8

The SU(3) zonal-characters χZ([1,1]) is obtained by SU(3) reduction from the 3-variables zonal polynomial ZP[{2,1,0}].

ZonalStructureConstantsResults_9.png

ZonalStructureConstantsResults_10.png

The notation [1,1] denotes the components, in the Dynkin basis (basis of fundamental weights), of the highest weight of the adjoint representation of SU(3).
The corresponding integer partition (Young diagram) is {2,1} and the corresponding extended partition (three components, with one trailing 0) is {2,1,0}.

We now consider the square of χZ([1,1])  and decompose on the the χZ([ν1,ν2]).
One gets

ZonalStructureConstantsResults_11.png

It is obtained by SU(3) reduction from the square of the zonal polynomial ZP[{2,1,0}], using the P normalization (hence ZP).

We consider the square of [s,s]  for s = 1, 2, 3, ...
Arguments { , } below (in this subsection) denote highest weights [ν1,ν2] in the SU(3) Dynkin basis.
The last entry m of each pair {{ν1,ν2},m] is the structure constant for the triple :  [s,s]  *  [s,s]  --> [ν1,ν2].

ZonalStructureConstantsResults_12.png

ZonalStructureConstantsResults_13.png

ZonalStructureConstantsResults_14.png

ZonalStructureConstantsResults_15.png

ZonalStructureConstantsResults_16.png

ZonalStructureConstantsResults_17.png

ZonalStructureConstantsResults_18.png

ZonalStructureConstantsResults_19.png

History and citation

The package SymPol$Package was written by R.C. during the academic year 2017/2018.
It was made public on the R.C. web site in September 2018.
If you use the SymPol$Package or the results obtained with it (for instance the results given above) in a scientific publication or talk, please give proper academic credit to the author.
Thank you.

Created with the Wolfram Language