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51
Cluster algebras as Hall algebras of quiver representations
"... Abstract. Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the ..."
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Cited by 60 (3 self)
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Abstract. Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the
Cluster mutation via quiver representations
 Comment. Math. Helv
"... Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of ..."
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Cited by 45 (15 self)
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Abstract. Matrix mutation appears in the definition of cluster algebras of Fomin and Zelevinsky. We give a representation theoretic interpretation of matrix mutation, using tilting theory in cluster categories of hereditary algebras. Using this, we obtain a representation theoretic interpretation of cluster mutation in case of acyclic cluster algebras.
Semicanonical bases and preprojective algebras
 Ann. Sci. École Norm. Sup
"... Abstract. Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynk ..."
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Cited by 34 (7 self)
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Abstract. Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as n. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky’s theory of cluster algebras. It was inspired by recent results of Caldero and Keller. 1. Introduction and
Generalized cluster complexes and Coxeter combinatorics
 Int. Math. Res. Notices
"... and study a simplicial complex ∆m (Φ) associated to a finite root system Φ and a nonnegative integer parameter m. Form = 1, our construction specializes to the (simplicial) ..."
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Cited by 23 (1 self)
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and study a simplicial complex ∆m (Φ) associated to a finite root system Φ and a nonnegative integer parameter m. Form = 1, our construction specializes to the (simplicial)
Noncrossing partitions in surprising locations
"... Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser know ..."
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Cited by 23 (1 self)
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Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples exhibiting this intrusive type of behavior include the Fibonacci numbers, the Catalan numbers, the quaternions, and the modular group. In this article, the focus is on a lesser known example: the noncrossing
Cluster algebras and triangulated surfaces. Part I: Cluster complexes
"... Abstract. We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of ..."
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Cited by 19 (1 self)
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Abstract. We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of “tagged triangulations” of the surface, and determine its homotopy type and its growth rate. Contents
Cambrian Fans
"... Abstract. For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc betwee ..."
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Cited by 19 (5 self)
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Abstract. For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc between csortable elements and cclusters induces a combinatorial isomorphism of fans. In particular, the cCambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the cCambrian fan are generated by certain vectors in the Worbit of the fundamental weights, while the rays of the ccluster fan are generated by certain roots. For particular (“bipartite”) choices of c, we show that the cCambrian fan is linearly isomorphic to the ccluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on cclusters by the cCambrian lattice. We give a simple bijection from cclusters to cnoncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric
Realizations of the associahedron and cyclohedron, preprint math.CO/0510614
"... Abstract. We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the BottTaubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which ..."
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Cited by 18 (4 self)
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Abstract. We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the BottTaubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type An or Bn respectively as only input and which specialises to a procedure presented by J.L. Loday for a certain orientation of An. 1.
Shellability of noncrossing partition lattices
 Proc. Amer. Math. Soc
"... Abstract. We give a casefree proof that the lattice of noncrossing partitions associated to any finite real reflection group is ELshellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three. 1. ..."
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Cited by 17 (3 self)
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Abstract. We give a casefree proof that the lattice of noncrossing partitions associated to any finite real reflection group is ELshellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three. 1.
Cluster algebras: Notes for the CDM03 conference
"... Abstract. This is an expanded version of the notes of our lectures ..."
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Cited by 17 (6 self)
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Abstract. This is an expanded version of the notes of our lectures