Symbolic Interpretation of the Molien Function: Free and Non-free Modules of Covariants

Abstract

A mathematical problem originating from molecular physics leads to the exploration of the algebraic structures of sets of multivariate polynomials whose variables are the (x i, y i) components of n vectors in a plane with a common origin. The symmetry is assumed to be described by the SO(2) Lie group. The irreducible representations (irreps) of the group are labeled by the integer m. The ring of invariants is the set of polynomials that transform under the action of the SO(2) group according to the m = 0 irrep. Such a ring admits a Cohen–Macaulay decomposition. The set of polynomials changing as the (m) irrep, m ≠ 0, under the elements of the group defines the module of (m)-covariants. The module of (m)-covariants is free when |m| < n and the expression of the Molien function is symbolically interpreted in terms of a standard integrity basis containing one set of denominator polynomials and one set of numerator polynomials. In contrast, the module of (m)-covariants is non-free when |m| ≥ n and a generalized integrity basis has to be introduced to throw light on the Molien function. A graphical representation of the algebraic structures of the free and non-free modules is proposed.