Additive Vs monoidal categorification of cluster algebras

In this talk, we will show how representations of a (Dynkin) quiver allow to construct cluster variables for the associated Fomin-Zelevinsky cluster algebra. We will start from the basics on quiver representations, notably Gabriel's theorem establishing a bijection between positive roots and indecomposable representations. By combining this bijection with Fomin-Zelevinsky's between positive roots and non-initial cluster variables, we will obtain a map associating a non-initial cluster variable with each indecomposable representation. The starting point of additive categorification is an explicit formula for this map due to Caldero-Chapoton. It involves Euler characteristics of varieties of subrepresentations and is typical of links between cluster algebras and algebraic geometry.