## Tanguy Vernet [1]

Quivers are finite graphs with an orientation. Like groups or Lie algebras, they have a representation theory and a natural goal is then to classify their indecomposable representations.

In this talk, I will cover foundational results concerning the classification of indecomposable quiver representations and illustrate them on elementary examples. The culminating point is Kac's theorem (1982), which links indecomposable representations to root systems of Lie algebras.

The proof relies on a geometric interpretation, in which isomorphism classes of representations are identified to orbits of some affine space under the action of an algebraic group. It involves arithmetic techniques and a polynomial counting of representations over finite fields (Kac's polynomial). It was proved more recently that Kac's polynomial has non-negative coefficients (Hausel, Letellier, Rodriguez-Villegas 2013). I will try to give some intuition for the proof in a particular case (Crawley-Boevey, Van den Bergh 2004). If time allows, I will say a few words about a related polynomial counting over finite quotients of local fields.