On the quiver Grassmannian in the acyclic case

By Philippe Caldero, Markus Reineke. 2006

123 citations recorded by the source corpus.

Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/26d7d3e89762f6d55370479c98208a01d364eb31

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Citations in the corpus

Tensor diagrams and cluster algebras (2012, background): “This conjecture has been proved in many special cases, see in particular [3, 41, 42].”
Local models of Shimura varieties, I. Geometry and combinatorics (2010, background): “In [14] and other papers in the area of representations of algebras, quiver Grassmannians are considered as varieties, i.”
Acyclic Calabi–Yau categories (2006, background): “It was shown [BMRRT06, BMR06, BMR08, BMRT05, CC06, CK05, CK06] that this category fully determines the combinatorics of the cluster algebra associated with Q and carries considerably more information.”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “exchange triangles xx∗ = m+m′ Tk →M →T∗ k →ΣTk T∗ k →M ′ →T k →ΣT∗ k 6.2. Two applications. Theorem6.1 doesshed new light oncluster algebras. In particular, thanks to the theorem, Caldero and Reineke [31] have made signiﬁcant progress towards the Conjecture 6.2. Suppose that Q does not have oriented”
Cluster algebras and quantum affine algebras (2009, background): “y of cluster expansions or linear independence of cluster monomials in some cases, this always requires some additional work, for example to show the positivity of some Euler characteristics (see e.g.[CR]). Finally, to recover the cluster algebra from its additive categoriﬁcation one does not consid”
Cluster categories (2010, background): “In this case there is the following information on the numerators of the cluster variables, when they are expressed in terms of the cluster in a seed which has an acyclic quiver ([35],[36],[98]).”
On a Quantum Analog of the Caldero–Chapoton Formula (2010, background): “When all valuations of the quiver Q are equal it is known, see [5], that for an indecomposable exceptional representation M we have |Gre(M)||F|→1 = χc(Gre(M)).”
Quiver varieties and cluster algebras (2009, background): “The dual of the subalgebra is a quotient.”
Quivers with potentials and their representations II: Applications to cluster algebras (2009, background): “It was shown in [7, 8], 2 that the F -polynomial of M has nonnegative coeﬃcients in this case.”
Quiver Grassmannians and degenerate flag varieties (2011, methodology): “Subsequently, specific classes of quiver Grassmannians (for example, varieties of subrepresentations of exceptional quiver representations) were studied by several authors, with the principal aim of computing their Euler characteristic explicitly; see for example [5, 6, 7, 8].”