The action of the orthogonal group on planar vectors: Invariants, covariants and syzygies

摘要

The construction of invariant and covariant polynomials from the x, y components of n planar vectors under the SO(2) and O(2) orthogonal groups is addressed. Molien functions determined under the SO(2) symmetry group are used as a guide to propose integrity bases for the algebra of invariants and the modules of covariants. The Molien functions that describe the structure of the algebra of invariants and the free modules of (m)-covariants, m n - 1, are written as a ratio of a numerator in λ with positive coefficients over a (1 - λ~2)~(2n - 1) denominator. This form of single rational function is standard in invariant theory and has a clear symbolic interpretation. However, its usefulness is lost for the non-free modules of (m)-covariants, m n, due to negative coefficients in the numerator. We propose for these non-free modules a new representation of the Molien function as a sum of n rational functions with positive coefficients in the numerators and different numbers of terms in the denominators. This non-standard form is symbolically interpreted in terms of a generalized integrity basis. Integrity bases are explicitly given for n = 2, 3, 4 planar vectors and m ranging from 0 to 5. The integrity bases obtained under the SO(2) symmetry group are subsequently extended to the O(2) group.