Symmetries of independent statistical observables for ultrametric populations

When N = 2(G), random data X-i, i = 1,...,N show ultrametric covariations, represented by a binary tree, decorrelated observables are defined by a covariance matrix diagonalization. Eigenvalue degeneracies lend one to regroup such observables into eigenprojectors. Symmetries of such projectors are discussed. Such symmetries and degeneracies influence the robustness of the corresponding measurements under random permutations of data.