Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors

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

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

A method for the construction of generalized integrity bases of invariant and covariant polynomials built from a set of three dimensional vectors under the SO(3) and O(3) symmetry is presented. The work is a follow–up to our previous article [G. Dhont and B. I. Zhilinskiı́ 2013. The action of the orthogonal group on planar vectors: invariants, covariants and syzygies. J. Phys. A: Math. Theor., 46, 455202] that dealt with a set of two dimensional vectors under the action of the SO(2) and O(2) groups. The Molien generating functions for invariant rings and covariant modules under SO(3) are derived. Their expressions as one rational function are a useful guide to build integrity bases for rings of invariants and free modules of covariants. We show how to deal with the Molien generating functions for non–free modules of covariants and how to take advantage of their expressions as a sum of rational functions to obtain generalized integrity bases. The O(3) invariants and covariants bases are deduced from the SO(3) ones. The results can be used in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are useful for the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time.

Mots clés: orthogonal group; invariant theory; integrity basis; free module; non-free module; dipole moment; quadrupole moment; orthogonal group

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

Guillaume Dhont, Patrick Cassam-Chenaï, Frédéric Patras. Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors. 2021. ⟨hal-01222972v3⟩