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

Abstract : The present article completes the mathematical description initiated in the paper by Dhont and Zhilinskií (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 $\left( m \right)$-covariant polynomials contains all the polynomials that transform according to the weight $m\in \mathbb{Z}$ of SO(2) and is a free module for $|m|\leqslant n-1$ but a non-free module for $|m|\geqslant n$. The sum of the rational functions of the Molien function for $\left( m \right)$-covariants describes the decomposition of the ring of invariants or the module of $\left( m \right)$-covariants as a direct sum of submodules. A method for extracting the generating function for $\left( m \right)$-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 $\left( m \right)$-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.
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Submitted on : Friday, May 29, 2015 - 3:54:50 PM
Last modification on : Friday, January 12, 2018 - 1:49:22 AM

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Guillaume Dhont, Frédéric Patras, Boris I Zhilinskií. The action of the special orthogonal group on planar vectors: integrity bases via a generalization of the symbolic interpretation of Molien functions. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2015, 48 (3), pp.035201. ⟨10.1088/1751-8113/48/3/035201⟩. ⟨hal-01158118⟩

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