Effective rationnality of second order symmetric tensor spaces

Marc Olive (1) , Rodrigue Desmorat (1)

1 LMT - Laboratoire de mécanique et technologie

We consider the natural SO(3, k) linear representation, k = C or R, on a k vector space of n second order symmetric tensors, the field of invariants 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 field of invariants 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 obtain last a minimal set of generators for the SL(2, C)-rational invariant field of n binary quartics.

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]

Marc Olive, Rodrigue Desmorat. Effective rationnality of second order symmetric tensor spaces. 2020. ⟨hal-03081027⟩