The lattice of submodules of a multiplicity free module

In this paper we determine, under some mild restrictions, the lattice of submodules $\text{gL}$ of a module $M$ all of whose composition factors have multiplicity one. Such a lattice is distributive, and hence determined by its poset of down-sets $P$. We define a directed Ext graph ${\text{Ext}}_{\text{gL}}$ of $\text{gL}$ and show that if ${\text{Ext}}_{\text{gL}}$ is acyclic, then ${\text{Ext}}_{\text{gL}}$ determines $P$. The result applies to multiplicity free indecomposable modules for finite dimensional algebras with acyclic Ext graph. It also applies to some deformed Verma modules which arise in the Jantzen sum formula basic classical simple Lie superalgebras in the deformed case.