The shift orbits of the graded Kronecker modules

TL;DR

This paper studies the structure and classification of graded Kronecker modules through the lens of shift orbits, revealing linear growth in radius and establishing a correspondence with paths in the n-regular tree.

Contribution

It introduces a new classification of regular graded Kronecker modules using shift orbits, minimal radius, and center paths, providing a coherent indexing method.

Findings

Radius of modules grows linearly under shift functor

Shift orbits are characterized by center paths and minimal radius

Correspondence between invariants and shift orbits established

Abstract

The Kronecker modules (or matrix pencils) are the representations of the n-Kronecker quiver K(n) (the quiver with two vertices, namely a sink and a source, and n arrows) over some fixed field. The universal cover of K(n) is the n-regular tree with bipartite orientation. The paper deals with the representations of the n-regular tree with bipartite orientation (thus with graded Kronecker modules). The simultaneous Bernstein-Gelfand-Ponomarev reflection at all sinks will be called the shift functor, and we consider the orbits under this shift functor. Whereas the length of a regular module growths exponentially when we apply the shift functor repeatedly, the radius of such a module growths just linearly. To any regular shift orbit, we attach a positive integer r (the minimal radius of the sink modules in the orbit) and the path in T(n) which starts at the center p of the sink modules in the orbit and ends at the center q of the source modules in the orbit. If r is even, then p has to be a sink, otherwise a source. We call this path the center path of the orbit (since the center of any module in the orbit lies on this path). If the center path of a shift orbit has length b, then the orbit contains precisely b flow modules, the remaining modules are sink modules and source modules. We use this division in order to index the regular graded Kronecker modules in a coherent way. Conversely, we show that given a positive integer r and a path in T(n) (starting at a sink iff r is even), there are regular shift orbits with these invariants.