Horn problem, hive polytopes and SU(n) Littlewood-Richardon coefficients

The notations J_n and A_j are those of [Zuber arXiv:1705.01186] and [Coquereaux-Zuber arXiv:1706.02793]:

In the generic case, the volume J_n of the SU(n) hive polytope associated with the triple (α, β ; γ) may be written as a double sum over two copies of the permutation group S_n of a quantity X_n,

X_n is the sum of the elements of the table Tn given below for n=4,5 and 6, in which A_j stands for the following linear form in α, β, γ

T4 (T4)

T5 (T5)

T6 (T6)

For example, compare the expression that is obtained for J₄ to that given in Appendix B.2 of [arXiv:1705.01186].