This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver Grassmannians. The notes are quite elementary and thought for phd students or young researchers. I assume that the reader is familiar with the representation theory of quivers.
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a
subvariety of a Kac-Moody group – the quiver is an orientation of its Dynkin diagram, defining a
Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of
the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective)
quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight)
The paper includes a new proof of the fact that quiver Grassmannians associated
with rigid representations of Dynkin quivers do not have cohomology in odd degrees.
Moreover, it is shown that they do not have torsion in homology. A new proof of the
Caldero-Chapoton formula is provided. As a consequence a new proof of the positivity of
cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The
methods used here are based on joint works with Markus Reineke and Evgeny Feigin.
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