  We discuss the diagonalization problem of the Nijenhuis tensor in a class of
Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We
study its action on the ring of invariant polynomials of a Thimm chain of
subalgebras. The existence of $\phi$-minimal representations defines a suitable
basis of invariant polynomials that completely solves the diagonalization
problem. We prove that such representations exist in the classical cases AIII,
BDI, DIII and CI, and do not exist in the exceptional cases EIII and EVII. We
discuss a second general construction that in these two cases computes
partially the spectrum and hints at a different behavior with respect to the
classical cases.
