Harder-Narasimhan filtrations of persistence modules (Q6658776)

The notion of semistability was introduced by Mumford more than 60 years ago to construct well-behaved quotient spaces for the actions of reductive groups on algebraic varieties. Subsequently, Harder and Narasimhan described a stratification of the moduli space of finite rank vector bundles over a complex curve for the purpose of computing its cohomology groups. The top stratum consists of semistable bundles, and every bundle lying in a lower stratum admits a canonical filtration of finite length whose associated graded components are semistable with strictly decreasing slopes (i.e., ratios of degree to rank). This Harder-Narasimhan filtration still plays a vital role in moduli problems involving vector bundles and coherent sheaves.\N\NThe Harder-Narasimhan type of a quiver representation is a discrete invariant parameterized by a real-valued function (called a central charge) defined on the vertices of the quiver. M. Fersztand, E. Jacquard, V. Nanda, and U. Tillmann investigate the strength and limitations of Harder-Narasimhan types for several families of quiver representations which arise in the study of persistence modules. They introduce the skyscraper invariant, which amalgamates the Harder-Narasimhan types along central charges supported at single vertices, and generalize the rank invariant from multi-parameter persistence modules to arbitrary quiver representations.\N\NThe authors show that the skyscraper invariant is strictly finer than the rank invariant and incomparable to the generalized rank invariant. They characterize the set of complete central charges for zigzag (and hence, ordinary) persistence modules, and then they extend the preceding characterization to rectangle-decomposable multi-parameter persistence modules of arbitrary dimension. Finally, the authors show that although no single central charge is complete for interval-decomposable ladder persistence modules, a finite set of central charges is complete.