Algorithms for fundamental invariants and equivariants
Résumé
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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