The action of the orthogonal group on planar vectors: invariants, covariants and syzygies (Q2866985)

As is known, the displacement vectors of vibrational modes of any quasi-rigid linear molecule with \(n\) atoms enter as basic elements the construction of invariants, such as e.g. the effective Hamiltonian or covariants, such as e.g. the electric dipole moment. The theory of invariants is a branch of mathematics based on group theory and algebra that provides a way to construct these polynomials, alternatively to the typical ones by means of a lengthy step-by-step approach of constructing all possible terms of degree \(n\) compatible with the final representation of the symmetry group from simpler terms of lower degree. This work addresses the problem of constructing invariant and covariant polynomials of \(x\) and \(y\) components of \(n\) planar vectors under the SO(2) and O(2) orthogonal groups. It invokes the Molien functions which are determined under the SO(2) symmetry group as a guide to build integrity bases for the algebra of invariants and the modules of covariants. It is demonstrated in this work that the Molien function for the ring of polynomial invariants and for the modules of polynomial \((m)\)-covariants built from the pair of components \((x,y)\) of one planar vector can be written as a single rational function that admits a symbolic interpretation in terms of integrity bases whose usefulness is however lost for the non-free modules of \((m)\)-covariants, \( m \geq n\), due to negative coefficients in the numerator. For the latter modules, a new representation of the Molien function is proposed. This representation is symbolically interpreted in terms of generalized integrity bases which are explicitly given for \(n=2,3,4\) planar vectors and \(m\) ranging from 0 to 5.