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Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras. (English) Zbl 1267.13043

The article under review studies quiver Grassmannians associated with finite dimensional indecomposable representations of a Kronecker quiver. A finite dimensional representation of a Kronecker quiver on the category of vector spaces is given by two finite dimensional complex vector spaces, \(M_{1}\) and \(M_{2}\) and linear maps \(m_{a}\), \(m_{b}\) between them. Given \(e_{1}\) and \(e_{2}\), define \[ Gr_{(e_{1},e_{2})}=\{(N_{1},N_{2})\in Gr_{e_{1}}(M_{1})\times Gr_{e_{2}}(M_{2}):m_{a}(N_{1})\subset M_{2},m_{b}(N_{1})\subset N_{2}\} \] which is a projective variety, in general non smooth, where \(Gr_{e}(V)\) denotes the Grassmannian of \(e\)-dimensional vector subspaces of \(V\). We call these varieties quiver Grassmannians.
For the case \(M_{1}=M_{2}=\mathbb{C}^{n}\), \(m_{a}=Id\) and \(m_{b}\) an indecomposable nilponent Jordan block, the representation is denoted by \(R_{n}\) and is called regular indecomposable. The article concentrates on this particular case, \(X=Gr_{(e_{1},e_{2})}(R_{n})\), because all the rest of indecomposable representations either have the same quiver, or are easier cases previously studied in the literature.
There is an action of a \(1\)-dimensional torus on \(X\) which provides a stratification \[ X_{s}\subseteq\cdots\subseteq X_{1}\subseteq X_{0}=X \] into closed subvarieties \(X_{k}\simeq Gr_{(e_{1}-k,e_{2}-k)}(R_{n-2k})\), where each one is the singular locus of the previous one and the difference between two consecutive ones is smooth quasiprojective. Bialynicki-Birula’s theorem is applied to show that \(X\) has a cellular decomposition and, relations between the dimensions of the cells through the Euler form associated to the Kronecker quiver, are used to compute the Betti numbers and the Poincare polynomials of the quiver Grassmannians, from which the Euler characteristics are recovered.
As an application, the authors give a new geometric realization for the atomic basis of a cluster algebra \(\mathcal{A}_{Q}\) of type \(A_{1}^{(1)}\) (which are \(\mathbb{Z}\)-subalgebras of the field of rational functions associated to quivers \(Q\) without loops and \(2\)-cycles) by using the Caldero-Chapoton map (cf. [P. Caldero and F. Chapoton, Comment. Math. Helv. 81, No. 3, 595–616 (2006; Zbl 1119.16013)]) associating cluster monomials of the cluster algebra to rigid \(Q\)-representations. The authors propose a truncation of the Caldero-Chapoton map to realize the extra elements of the atomic basis.

MSC:

13F60 Cluster algebras
06B15 Representation theory of lattices
16G20 Representations of quivers and partially ordered sets
14M15 Grassmannians, Schubert varieties, flag manifolds
05E10 Combinatorial aspects of representation theory
14N05 Projective techniques in algebraic geometry

Citations:

Zbl 1119.16013
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