Non-semistable exceptional objects in hereditary categories: some remarks and conjectures

In our previous paper we studied non-semistable exceptional objects in hereditary categories and introduced the notion of regularity preserving category, but we obtained quite a few examples of such categories. Certain conditions on the Ext-nontrivial couples (exceptional objects $X,Y\in \mathcal A$ with ${\rm Ext}^1(X,Y)\neq 0$ and ${\rm Ext}^1(Y,X)\neq 0$) were shown to imply regularity-preserving. This paper is a brief review of the previous paper (with emphasis on regularity preserving property) and we add some remarks and conjectures. It is known that in Dynkin quivers ${\rm Hom}(\rho,\rho')=0$ or ${\rm Ext}^1(\rho,\rho')=0$ for any two exceptional representations. On one hand, in the present paper we prove this fact by a new method, which allows us to extend it to representation infinite cases: the extended Dynkin quivers $\widetilde{\mathbb E}_6, \widetilde{\mathbb E}_7, \widetilde{\mathbb E}_8 $. On the other hand, we use it to show that for any Dynkin quiver $Q$ there are no Ext-nontrivial couples in $Rep_k(Q)$, which implies regularity preserving of $Rep_k(Q)$, where $k$ is an algebraically closed field.