Regular modules with preprojective Gabriel-Roiter submodules over nn-Kronecker quivers

Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposable regular modules with preprojective Gabriel-Roiter submodules, in particular, those $\tau^{-i}X$ with $im X=(1,c)$ for $i\geq 0$ and some $1\leq c\leq n-1$ will be studied. It will be shown that for each $i\geq 0$ the irreducible monomorphisms starting with $\tau^{-i}X$ give rise to a sequence of Gabriel-Roiter inclusions, and moreover, the Gabriel-Roiter measures of those produce a sequence of direct successors. In particular, there are infinitely many GR-segments, i.e., a sequence of Gabriel-Roiter measures closed under direct successors and predecessors. The case $n=3$ will be studied in detail with the help of Fibonacci numbers. It will be proved that for a regular component containing some indecomposable module with dimension vector $(1,1)$ or $(1,2)$, the Gabriel-Roiter measures of the indecomposable modules are uniquely determined by their dimension vectors

The number of the Gabriel-Roiter measures admitting no direct predecessors over a wild quiver

A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel-Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel-Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of artin algebras. As the first step, we are going to study the wild quiver with three vertices, labeled by $1$,$2$ and $3$, and one arrow from $1$ to $2$ and two arrows from $2$ to $3$. The Gabriel-Roiter submodules of the indecomposable preprojective modules and quasi-simple modules $\tau^{-i}M$, $i\geq 0$ are described, where $M$ is a Kronecker module and $\tau=DTr$ is the Auslander-Reiten translation. Based on these calculations, the existence of infinitely many GR segments will be shown. Moreover, it will be proved that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors