Do there exist methods for determining the orbits of a group action on the cartesian product of sets?

Suppose that we have some group $G$ acting on some set $\Omega$. Then $G$ acts on $\Omega^n = \Omega \times \cdots \times \Omega$ ($n$ times) naturally. I wonder, is there an iterative algorithm to determine orbit representatives for the action of $G$ on $\Omega^n$ using information from the action of $G$ on $\Omega$? Of course, there are algorithms to determine orbit representatives from any group action, but since the action of $G$ on $\Omega^n$ relates naturally to the action $G$ on $\Omega$, I hoped that these orbits could be computed more efficiently than on some arbitrary set.

Please note that I am not asking about computing the number of said orbits; I know there are questions here that already address that. I am looking for representatives.

You get representatives $(\omega_1,\ldots,\omega_n)$ of orbits on tuples by:

• Choosing the $\omega_1$ as orbit representatives under $G$.
• For each $\omega_1$, choose $\omega_2$ as orbit representatives under $Stab_G(\omega_1)$.
• For each pair $(\omega_1,\omega_2)$, choose $\omega_3$ as orbit representatives under $Stab_G(\omega_1,\omega_2)$.

and so on. This is probably easiest implemented recursively. (The proof of this is straightforward -- every element can be brought into such a form under $G$ and no two elements are equivalent, as otherwise some $\omega_i$ would be in the same orbit.)

If you want orbit representatives on sets, the problem becomes much harder:

You can of course use tuple representatives as first approximation and use some lexicography (representative is smallest under entry permutation) to discard obvious conjugates.

Ultimately you will need to test for set equivalence. The paper https://doi.org/10.1145/1005285.1005319 is probably a good initial read.

Finally, the particular choice of groups might allow for further restrictions in the construction tree, but this depends on the group in question.