The discriminating power of the generalized rank invariant

It is a well-known fact that there is no complete and discrete invariant on the collection of all multiparameter persistence modules. Nonetheless, many invariants have been proposed in the literature to study multiparameter persistence modules, though each invariant will lose some amount of information. One such invariant is the generalized rank invariant. This invariant is known to be complete on the class of interval decomposable persistence modules in general, under mild assumptions on the indexing poset $P$. There is often a trade-off, where the stronger an invariant is, the more expensive it is to compute in practice. The generalized rank invariant on its own is difficult to compute, whereas the standard rank invariant is readily computable through software implementations such as RIVET. We can interpolate between these two to induce new invariants via restricting the domain of the generalized rank invariant, and this family exhibits the aforementioned trade-off. This work studies the tension which exists between computational efficiency and retaining strength when restricting the domain of the generalized rank invariant. We provide a characterization result on where such restrictions are complete invariants in the setting where $P$ is finite, and furthermore show that such restricted generalized rank invariants are stable.