Let Q be an orientation of a type A Dynkin diagram. An eta map corresponding to Q is a surjection from the weak order on permutations to a Cambrian lattice (of triangulations of a polygon). We give a new geometric way to construct the Auslander-Reiten quiver of the quiver representations rep(Q). We use it to naturally define a quiver representation which is a (new) Catalan object, and we define an analog of the eta map onto the set of these representations in rep(Q).

This is a preliminary report on joint work with E. Barnard, E. Meehan, and R. Schiffler.