Equivariant subspaces of orbit-finite-dimensional vector spaces

In their LICS21 paper, Bojańczyk, Klin, and Moerman gave an equivalence algorithm for weighted orbit-finite automata: they construct a chain of configuration spaces that are closed under both linear combinations and atom permutations; crucially, termination is guaranteed by a finite length property for such chains of equivariant subspaces. Towards understanding the full subspace structure of vector spaces with an infinite but orbit-finite basis, we describe how finite-dimensional representation theory can help.