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Positivity for cluster algebras from surfaces
, 2009
"... We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our for ..."
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Cited by 43 (11 self)
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We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.
On cluster algebras arising from unpunctured surfaces II
, 2008
"... We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an im ..."
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Cited by 30 (11 self)
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We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for Fpolynomials and gvectors and show that Fpolynomials have constant term equal to 1. As an application, we compute the EulerPoincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type Ã.
Cluster expansion formulas and perfect matchings
 J. ALGEBRAIC COMBIN
, 2008
"... We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph GT,γ that is constructed from the ..."
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Cited by 20 (8 self)
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We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ.
GENERIC BASES FOR CLUSTER ALGEBRAS FROM THE CLUSTER CATEGORY
"... Abstract. Inspired by recent work of Geiss–Leclerc–Schröer, we use Homfinite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same i ..."
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Cited by 15 (0 self)
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Abstract. Inspired by recent work of Geiss–Leclerc–Schröer, we use Homfinite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss–Leclerc–Schröer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock–Goncharov concerning the parametrization of a basis of the upper cluster
Coloured quivers for rigid objects and partial triangulations: the unpunctured case
 Proc. Lond. Math. Soc
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On the cluster category of a marked surface without punctures
 ALGEBRA AND NUMBER THEORY 5:4(2011)
, 2011
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