The orthogonal group action on spatial vectors: invariants, covariants, and syzygies

Guillaume Dhont (1) , Patrick Cassam-Chenaï (2) , Boris Zhilinskii (1) , Frédéric Patras (2)

1 LPCA - Laboratoire de Physico-Chimie de l'Atmosphère
2 JAD - Laboratoire Jean Alexandre Dieudonné

The present paper on the SO(3) invariants and covariants built from N vectors of the three–dimensional space is the follow-up of our previous article [1] dealing with planar vectors and SO(2) symmetry. The goal is to propose integrity basis for the set of SO(3) invariants and covariant free modules and easy-to-use generating families in the case of non-free covariants modules. The existence of such non-free modules is one of the noteworthy features unseen when dealing with finite point groups, that we want to point out. As in paper [1], the Molien function plays a central role in the conception of these bases. The Molien functions are computed and checked by the use of two independent paths. The first computation relies on the Molien integral [2] and requires the matrix representation of the group action on the N spatial vectors. The second path considers the Molien function for only one spatial vector as the elementary building material from which are worked out the other Molien functions.

Domaines: Algèbre commutative [math.AC]; Physique mathématique [math-ph]

Guillaume Dhont, Patrick Cassam-Chenaï, Boris Zhilinskii, Frédéric Patras. The orthogonal group action on spatial vectors: invariants, covariants, and syzygies. 2015. ⟨hal-01222972v1⟩