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Tilting theory and cluster combinatorics
Abstract We introduce a new category C , which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebraExpand
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Cluster-tilted algebras
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theoryExpand
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Parametrizations of flag varieties
For the flag variety G/B of a reductive algebraic group G we define and describe explicitly a certain (set-theoretical) cross-section φ : G/B → G. The definition of φ depends only on a choice ofExpand
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Cluster mutation-periodic quivers and associated Laurent sequences
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all theExpand
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CLUSTER MUTATION VIA QUIVER REPRESENTATIONS
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 categoriesExpand
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A geometric description of m-cluster categories
We show that the m-cluster category of type An 1 is equivalent to a certain geometrically-defined category of diagonals of a regular nm + 2-gon. This generalises a result of Caldero, Chapoton andExpand
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Reflection group presentations arising from cluster algebras
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms ofExpand
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Cluster-tilted algebras of finite representation type
Abstract We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finiteExpand
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Twists of Plücker Coordinates as Dimer Partition Functions
The homogeneous coordinate ring of the Grassmannian Grk,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consistsExpand
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Braid groups and quiver mutation
We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A andExpand
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