Distributive lattices of tilting modules and support $τ$-tilting modules over path algebras

Abstract: In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $\tau$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}{1}, \mathbb{A}{2}$ or $\mathbb{A}{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}{1}$.