Nijenhuis tensor and invariant polynomials

Authors: F. Bonechi, J. Qiu, M. Tarlini, E. Viviani Preprint number: UUITP-54/21 Abstract: We discuss the diagonalization problem of the Nijenhuis tensor in a class of {\it 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 $\phi$-{\it minimal representations} defines a suitable basis of invariant polynomials that completely solves the diagonalization problem. We prove that such reprentations 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.