Effective rationality of second order symmetric tensor spaces

We consider the natural SO(3, k) linear representation, k = C or R, on the k vector space V = nS2(R3) of n second order symmetric tensors, the associated invariant field k(V)SO(3,k) being known to be a purely transcendental extension in the complex case. We give an explicit tensorial form of a minimal generating set of the field of invariants, in both the complex and the real cases, showing that the invariant field is also a purely transcendental extension in the real case. Present results rely on some octahedral polynomial invariants obtained from Clebsch-Gordan projectors defined by a fourth order octahedral covariant. Thanks to Cartan’s map we also obtain a minimal set of generators for the SL(2, C)-rational invariant field of n binary quartics.