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193
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 140 (5 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
Ysystems and generalized associahedra
 Ann. of Math
"... Root systems and generalized associahedra 1 Root systems and generalized associahedra 3 ..."
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Cited by 100 (10 self)
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Root systems and generalized associahedra 1 Root systems and generalized associahedra 3
Surface group representations with maximal Toledo invariant
, 2006
"... Abstract. We develop the theory of maximal representations of the fundamental group π1(Σ) of a compact connected oriented surface Σ with boundary ∂Σ, into the isometry group of a Hermitian symmetric space X or, more generally, a group of Hermitian type G. For any homomorphism ρ: π1(Σ) → G, we defin ..."
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Cited by 77 (17 self)
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Abstract. We develop the theory of maximal representations of the fundamental group π1(Σ) of a compact connected oriented surface Σ with boundary ∂Σ, into the isometry group of a Hermitian symmetric space X or, more generally, a group of Hermitian type G. For any homomorphism ρ: π1(Σ) → G, we define the Toledo invariant T(Σ, ρ), a numerical invariant which is in general not a characteristic number, but which has both topological and analytical interpretations: we establish important properties, among which uniform boundedness on the representation variety Hom � π1(Σ), G � , additivity under connected sum of surfaces and congruence relations mod Z. We thus obtain information about the representation variety as well as striking geometric properties of the maximal representations, that is representations whose Toledo invariant achieves the maximum value: we show that maximal representations have discrete image, are faithful and completely reducible and they always preserve a maximal tube type subdomain of X. This extends to the case of a general Hermitian group some of the properties of the representations in
Quantum cluster algebras
, 2004
"... 2. Cluster algebras of geometric type 3 3. Compatible pairs 5 4. Quantum cluster algebras setup 8 ..."
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Cited by 71 (6 self)
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2. Cluster algebras of geometric type 3 3. Compatible pairs 5 4. Quantum cluster algebras setup 8
Cluster algebras and WeilPetersson forms
 Duke Math. J
"... Abstract. In the previous paper [GSV] we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea is that a relevant geometr ..."
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Cited by 64 (3 self)
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Abstract. In the previous paper [GSV] we have discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper we consider the case of a general matrix of transition exponents. Our leading idea is that a relevant geometric object in this case is a certain closed 2form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmüller space, in which case the above form coincides with the classical WeilPetersson symplectic form. 1.
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 63 (2 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
Anosov flows, surface groups and curves in projective space
"... In his beautiful paper [13], N. Hitchin studied the connected components of the space ..."
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Cited by 57 (4 self)
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In his beautiful paper [13], N. Hitchin studied the connected components of the space
Perfect matchings and the octahedron recurrence
 math.CO/0402452, 2004. André Henriques, Mathematisches Institut, Westfälische WilhelmsUniversität, Einsteinstr. 62, 48149
"... We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conje ..."
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Cited by 51 (1 self)
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We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky. 1