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Algorithms for fundamental invariants and equivariants: (of finite groups)

Evelyne Hubert 1, 2 Erick Rodriguez Bazan 1, 2 
1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , NKUA - National and Kapodistrian University of Athens
Abstract : For a finite group, we present three algorithms to compute a generating set of invariant simultaneously to generating sets of basic equivariants, i.e., equivariants for the irreducible representations of the group. The main novelty resides in the exploitation of the orthogonal complement of the ideal generated by invariants; Its symmetry adapted basis delivers the fundamental equivariants. Fundamental equivariants allow to assemble symmetry adapted bases of polynomial spaces of higher degrees, and these are essential ingredients in exploiting and preserving symmetry in computations. They appear within algebraic computation and beyond, in physics, chemistry and engineering. Our first construction applies solely to reflection groups and consists in applying symmetry preserving interpolation, as developed by the same authors, along an orbit in general position. The fundamental invariants can be read off the H-basis of the ideal of the orbit while the fundamental equivariants are obtained from a symmetry adapted basis of an invariant direct complement to this ideal in the polynomial ring. The second algorithm takes as input primary invariants and the output provides not only the secondary invariants but also free bases for the modules of basic equivariants. These are constructed as the components of a symmetry adapted basis of the orthogonal complement, in the polynomial ring, to the ideal generated by primary invariants. The third algorithm proceeds degree by degree, determining the fundamental invariants as forming a H-basis of the Hilbert ideal, i.e., the polynomial ideal generated by the invariants of positive degree. The fundamental equivariants are simultaneously computed degree by degree as the components of a symmetry adapted basis of the orthogonal complement of the Hilbert ideal.
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Submitted on : Wednesday, March 23, 2022 - 1:09:43 PM
Last modification on : Wednesday, July 13, 2022 - 9:14:00 AM

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Evelyne Hubert, Erick Rodriguez Bazan. Algorithms for fundamental invariants and equivariants: (of finite groups). Mathematics of Computation, American Mathematical Society, 2022, 91 (337), pp.2459-2488. ⟨10.1090/mcom/3749⟩. ⟨hal-03209117v4⟩

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