  The Harder-Narasimhan type of a quiver representation is a discrete invariant
parameterised by a real-valued function (called a central charge) defined on
the vertices of the quiver. In this paper, we investigate the strength and
limitations of Harder-Narasimhan types for several families of quiver
representations which arise in the study of persistence modules. We introduce
the skyscraper invariant, which amalgamates the HN types along central charges
supported at single vertices, and generalise the rank invariant from
multiparameter persistence modules to arbitrary quiver representations. Our
four main results are as follows: (1) we show that the skyscraper invariant is
strictly finer than the rank invariant in full generality, (2) we characterise
the set of complete central charges for zigzag (and hence, ordinary)
persistence modules, (3) we extend the preceding characterisation to
rectangle-decomposable multiparameter persistence modules of arbitrary
dimension; and finally, (4) we show that although no single central charge is
complete for nestfree ladder persistence modules, a finite set of central
charges is complete.
