We can "represent" a directed graph by assigning vector spaces to the vertices and linear maps to the edges, and identify two representations if they differ by a change of basis at some of the vertices. Some numbers associated to a representation, which are invariant under such base change, include dimensions of the spaces and ranks of the maps. In classifying the representations of a fixed graph Q, it would be nice to have as many of these numerical invariants as possible for Q.

For a certain class of directed graphs (trees with a unique sink), we can combinatorially construct lattices that parametrize all the invariants which "behave like" ranks. This will be demonstrated through examples, and the general procedure discussed, but probably no thorough proof will be given in this talk. The audience will only be assumed to know what rank of a linear map is, everything else will be defined during the talk.