The Caldero–Chapoton formula for cluster algebras

In this article, we state the necessary background for cluster algebras and quiver representations to formulate and prove a result of Caldero and Chapoton which gives a nice formula connecting the two subjects. In our exposition, we mostly follow the paper [CC], but we have tried to minimize the reliance of citing outside sources as much as possible. In particular, we avoid having to introduce cluster categories and quantized universal enveloping algebras. Of course, a completely self-contained account would take up far too many pages, so we have taken the liberty of assuming some results from cluster algebras and quiver representations, and a little bit of Auslander–Reiten theory. All of the necessary definitions and results are given in the first section. This article was written as a final paper for the MIT course 18.735 Topics in Algebra: Quivers in Representation Theory, which was taught by Travis Schedler during the spring 2009 semester.