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If $\Gamma$ is a finite group with action on two vector spaces $\mathbb R^n$ and $\mathbb R^m$ denoted by $\gamma_n$ and $\gamma_m$ respectively, the fundamental equivariants are the polynomials $f: \mathbb R^n \rightarrow \mathbb R^m$ that commute with $\Gamma$, i.e. $$\gamma_m \cdot f(x) = f(\gamma_n^{-1} \cdot x) \quad \forall x \in \mathbb R^n.$$

It it well known that this set is a finitely generated free module over the ring of primary invariants.

Is there any software available to compute a set of generators of the module of equivariants?

An algorithm is described in page 12 of this paper "Zeros of equivariant vector fields..." by P. Worfolk (citeseerx link, unrestricted) for instance in pseudo code, but surprisingly I was not able to find any implementation online in Singular, Magma or similar.

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  • $\begingroup$ I guess this set can be extracted from the set of generators of invariant ring (for which you have a choice of implementations) of the direct sum $\gamma_n\oplus\gamma_m$. $\endgroup$ – Dima Pasechnik Mar 21 '18 at 19:23
  • $\begingroup$ True, but that seems like a waste since that would correspond the projection of the invariants a small subspace. (subspace of polynomials that are linear in the new slack variables) $\endgroup$ – maroxe Mar 21 '18 at 19:33
  • $\begingroup$ Have you tried googling google.com.ua/… ? $\endgroup$ – user64494 Mar 21 '18 at 19:38
  • $\begingroup$ typically one would first compute the primary invariants, and you need them anyway. In practice these computations a very hard... $\endgroup$ – Dima Pasechnik Mar 21 '18 at 20:21
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The canonical reference appears to be Karin Gatermann's book:

Gatermann, Karin, Computer algebra methods for equivariant dynamical systems, Lecture Notes in Mathematics. 1728. Berlin: Springer. xv, 153 p. (2000). ZBL0944.65131.

In the book she frequently alludes to the Maple package symmetry which she uses to compute invariants and equivariants.

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  • $\begingroup$ Thanks, that is "almost" perfect. The only issue is that Maple is not free. Do you know of any alternative? $\endgroup$ – maroxe Mar 21 '18 at 22:47

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