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35
QUANTUM UNIPOTENT SUBGROUP AND DUAL CANONICAL BASIS
, 2010
"... ... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture w ..."
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... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras in [4] to the dual canonical basis B up. In particular, we prove that the quantum analogue Oq[N(w)] of C[N(w)] has the induced basis from B up, which contains quantum flag minors and satisfies a factorization property with respect to the ‘qcenter’ of Oq[N(w)]. This generalizes Caldero’s results [7, 8, 9] from ADE cases to an arbitary symmetrizable KacMoody Lie algebra.
Categorification of acyclic cluster algebras: an introduction
 IN THE PROCEEDINGS OF THE CONFERENCE ‘HIGHER STRUCTURES IN GEOMETRY AND PHYSICS 2007’, BIRKHÄUSER
"... This is a concise introduction to FominZelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster v ..."
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This is a concise introduction to FominZelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.
A COMPENDIUM ON THE CLUSTER ALGEBRA AND QUIVER PACKAGE IN Sage
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 65 (2011), ARTICLE B65D
, 2011
"... This is the compendium of the cluster algebra and quiver package for Sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. I ..."
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Cited by 7 (2 self)
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This is the compendium of the cluster algebra and quiver package for Sage. The purpose of this package is to provide a platform to work with cluster algebras in graduate courses and to further develop the theory by working on examples, by gathering data, and by exhibiting and testing conjectures. In this compendium, we include the relevant theory to introduce the reader to cluster algebras assuming no prior background. Throughout this compendium, we include examples that the user can run in the Sage notebook or command line, and then close with a detailed description of the data structures and methods in this package.
A quantum analogue of generic bases for affine cluster algebras
 Science China Mathematics.55 (2012
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