Etienne Ménard

Cluster algebras are not so easy to find in nature. To catch them we need either luck or the help of representation theory. Here we will see the second approach. We will first recall principal notions on cluster algebras, introduce the context of our problem (cluster algebra on the coordinate ring of unipotent maximal subgroups) and the tools used to bait the cluster algebra. By building a nice category of modules over the preprojective algebra, we will spot some cluster structure on it.

We will eventualy transport the cluster structure found in the category to relocate it in its homeland of coordinate ring via an application called "Cluster character" whose concrete computation will be explained as well as its multiplicative property.