Nijenhuis tensor and invariant polynomials

TL;DR

This paper investigates the diagonalization of the Nijenhuis tensor within Poisson-Nijenhuis structures on compact hermitian symmetric spaces, providing explicit solutions in classical cases and partial results in exceptional cases.

Contribution

It introduces the concept of $$-minimal representations for invariant polynomials, solving the diagonalization problem in classical cases and exploring alternative approaches for exceptional cases.

Findings

Existence of $$-minimal representations in classical cases AIII, BDI, DIII, CI.

Non-existence of such representations in exceptional cases EIII, EVII.

Partial spectrum computation and different behavior in exceptional cases.

Abstract

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.