The Gabriel-Roiter submodules of simple homogeneous modules

Abstract:

Let $\Lambda$ be a connected tame hereditary algebra over an algebraically closed field. We show that if $\Lambda =kQ$ is of type $\widetilde {\mathbb {A}}_n$, $\widetilde {\mathbb {D}}_n$, $\widetilde {\mathbb {E}}_6$ or $\widetilde {\mathbb {E}}_7$, then every Gabriel-Roiter submodule of a quasi-simple module of rank $1$ (i.e. a simple homogeneous module) has defect $-1$. In particular, any Gabriel-Roiter submodule of a simple homogeneous module yields a Kronecker pair, and thus induces a full exact embedding of the category $\operatorname {mod} k\widetilde {\mathbb {A}}_1$ into $\operatorname {mod}\Lambda$, where $\widetilde {\mathbb {A}}_1$ is the Kronecker quiver. Consequently, we obtain that all quasi-simple modules are Gabriel-Roiter factor modules.