The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions

摘要

The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskii (2013 The action of the orthogonal group on planar vectors: invariants, covariants and syzygies J. Phys. A: Math. Theor. 46 455202) of the algebraic structures that emerge from the symmetry-adapted polynomials in the (x(i), y(i)) coordinates of n planar vectors under the action of the SO(2) group. The set of (m)-covariant polynomials contains all the polynomials that transform according to the weight m is an element of Z of SO(2) and is a free module for |m| <= n-1 but a non-free module for |m| >= n. The sum of the rational functions of the Molien function for (m)-covariants describes the decomposition of the ring of invariants or the module of (m)-covariants as a direct sum of submodules. A method for extracting the generating function for (m)-covariants from the comprehensive generating function for all polynomials is introduced. The approach allows the direct construction of the integrity basis for the module of (m)-covariants decomposed as a direct sum of submodules and gives insight into the expressions for the Molien functions found in our earlier paper. In particular, a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed, where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed.