Regular modules with preprojective Gabriel–Roiter submodules over n-Kronecker quivers

Let Q be a wild n-Kronecker quiver, i.e., a quiver with two vertices, one source and one sink, and n≥3 arrows from the source to the sink. The indecomposable regular modules with preprojective Gabriel–Roiter submodules will be studied. The direct successors of the Gabriel–Roiter measures of these kinds of indecomposable modules will be discussed. In particular, it will be shown that 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 some particular regular components the Gabriel–Roiter measures of the indecomposable modules are uniquely determined by their dimension vectors.