Guillaume Dhont

Symbolic interpretation of the Molien function: free and non-free modules of covariants

A problem from molecular physics led us to the investigation of the algebraic structures of the polynomials generated from the (x_i,y_i) components of n vectors in a plane with a common origin. The symmetry group is assumed to be SO(2) and the irreducible representations (m) are labelled by the integer m. The ring of invariants (m=0) is Cohen-Macaulay. The module of covariants is free when |m|<n but is non-free if |m| is greater or equal than the number of vectors. We discuss the symbolic interpretation of the corresponding Molien functions in term of integrity bases and propose a graphical representation of the structure of the modules.