Theta functions and quiver Grassmannians

In this article, we use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams. With a notion of positive crossing of a path $\gamma$, we show that if $\gamma$ has positive crossing in the scattering diagram, then it goes in the opposite direction of the Auslander-Reiten quiver of $Q$. We then give the Hall algebra theta functions which recover the cluster character formula by the Euler characteristic map. At last, we define the Hall algebra broken lines and then are able to give the stratification of the quiver Grassmannians by the bending of the broken lines.