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GeometryofQuiverGrassmannians.PDF
Geometry of Quiver Grassmannians of Dynkin type
with applications to cluster algebras
Giovanni Cerulli Irelli∗
6 Abstract. The paper includes a new proof of the fact that quiver Grassmannians associ-
1 ated with rigid representations of Dynkin quivers do not have cohomology in odd degrees.
0 Moreover, it is shown that they do not have torsion in homology. A new proof of the
2 Caldero–Chapoton formula is provided. As a consequence a new proof of the positivity of
l cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The
u methods used here are based on joint works with Markus Reineke and Evgeny Feigin.
J
5
1 2010 Mathematics Subject Classification. Primary 16G70; Secondary 14N05.
Keywords. Quiver Grassmannians, Dynkin quivers, Cluster Algebras.
]
A
R Introduction
.
h
t Quiver Grassmannians are the natural generalization in the world of quivers of the
a
usual Grassmannians of linear subspaces: given a representation M of a quiver Q
m and a dimension vector e, the quiver Grassmannians Gr (M) parametrizes the sub-
e
[
representations of M of dimension vector e. The name was suggested by Zelevin-
2 sky, and it is nowadays commonly used. We say that a quiver Grassmannian is of
v Dynkin type, if it is associated with a complex representation of a Dynkin quiver.
9 My first aim in writing this paper was to survey recents results obtained in
3
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