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Ryo Fujita - One of the best experts on this subject based on the ideXlab platform.
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Geometric Realization of Dynkin Quiver Type Quantum Affine Schur–Weyl Duality
International Mathematics Research Notices, 2018Co-Authors: Ryo FujitaAbstract:For a Dynkin quiver $Q$ of type ADE and a sum $\beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's Geometric Realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $\beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.
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Geometric Realization of dynkin quiver type quantum affine schur weyl duality
arXiv: Representation Theory, 2018Co-Authors: Ryo FujitaAbstract:For a Dynkin quiver $Q$ of type ADE and a sum $\beta$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's Geometric Realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $\beta$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl duality functor. As a by-product, we verify a conjecture by Kang-Kashiwara-Kim on the simpleness of some poles of normalized R-matrices for any quiver $Q$ of type ADE.
Ralf Schiffler - One of the best experts on this subject based on the ideXlab platform.
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A Geometric Realization of socle-projective categories for posets of type $\mathbb{A}$
arXiv: Representation Theory, 2019Co-Authors: Ralf Schiffler, Robinson-julian SernaAbstract:This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type $\mathbb{A}$ with certain additional arrows. This class of posets is therefore called \emph{posets of type $\mathbb{A}$}. We then give a Geometric Realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type $\mathbb{A}$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the Realization of the cluster category of type $\mathbb{A}$ as the category of all diagonals by Caldero, Chapoton and the first author. We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.
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a Geometric Realization of socle projective categories for posets of type mathbb a
arXiv: Representation Theory, 2019Co-Authors: Ralf Schiffler, Robinson-julian SernaAbstract:This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type $\mathbb{A}$ with certain additional arrows. This class of posets is therefore called \emph{posets of type $\mathbb{A}$}. We then give a Geometric Realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type $\mathbb{A}$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the Realization of the cluster category of type $\mathbb{A}$ as the category of all diagonals by Caldero, Chapoton and the first author. We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.
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a Geometric model for cluster categories of type dn
Journal of Algebraic Combinatorics, 2008Co-Authors: Ralf SchifflerAbstract:We give a Geometric Realization of cluster categories of type D n using a polygon with n vertices and one puncture in its center as a model. In this Realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
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A Geometric model for cluster categories of type D _ n
Journal of Algebraic Combinatorics, 2008Co-Authors: Ralf SchifflerAbstract:We give a Geometric Realization of cluster categories of type D _ n using a polygon with n vertices and one puncture in its center as a model. In this Realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
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Quivers with relations arising from clusters (A_n case)
Trans. Amer. Math. Soc., 2006Co-Authors: Philippe Caldero, Frédéric Chapoton, Ralf SchifflerAbstract:Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic Realization and a Geometric Realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.
Hermund André Torkildsen - One of the best experts on this subject based on the ideXlab platform.
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A Geometric Realization of the m-cluster Category of Affine Type A
Communications in Algebra, 2015Co-Authors: Hermund André TorkildsenAbstract:We give a Geometric Realization of a subcategory of the m-cluster category 𝒞 m of type , by using (m + 2)-angulations of an annulus with p + q marked points. We also give a bijection between an equivalence class of (m + 2)-angulations and the mutation class of coloured quivers of type .
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A Geometric Realization of tame categories
arXiv: Representation Theory, 2015Co-Authors: Karin Baur, Hermund André TorkildsenAbstract:We give a Geometric Realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the Geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
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A Geometric Realization of the $m$-cluster category of type $\tilde{A}$
arXiv: Representation Theory, 2012Co-Authors: Hermund André TorkildsenAbstract:We give a Geometric Realization of a subcategory of the $m$-cluster category $\mathcal{C}^m$ of type $\widetilde{A}_{p,q}$, by using $(m+2)$-angulations of an annulus with $p+q$ marked points. We also give a bijection between an equivalence class of $(m+2)$-angulations and the mutation class of coloured quivers of type $\widetilde{A}_{p,q}$.
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a Geometric Realization of the m cluster category of type tilde a
arXiv: Representation Theory, 2012Co-Authors: Hermund André TorkildsenAbstract:We give a Geometric Realization of a subcategory of the $m$-cluster category $\mathcal{C}^m$ of type $\widetilde{A}_{p,q}$, by using $(m+2)$-angulations of an annulus with $p+q$ marked points. We also give a bijection between an equivalence class of $(m+2)$-angulations and the mutation class of coloured quivers of type $\widetilde{A}_{p,q}$.
Robinson-julian Serna - One of the best experts on this subject based on the ideXlab platform.
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a Geometric Realization of socle projective categories for posets of type mathbb a
arXiv: Representation Theory, 2019Co-Authors: Ralf Schiffler, Robinson-julian SernaAbstract:This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type $\mathbb{A}$ with certain additional arrows. This class of posets is therefore called \emph{posets of type $\mathbb{A}$}. We then give a Geometric Realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type $\mathbb{A}$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the Realization of the cluster category of type $\mathbb{A}$ as the category of all diagonals by Caldero, Chapoton and the first author. We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.
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A Geometric Realization of socle-projective categories for posets of type $\mathbb{A}$
arXiv: Representation Theory, 2019Co-Authors: Ralf Schiffler, Robinson-julian SernaAbstract:This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type $\mathbb{A}$ with certain additional arrows. This class of posets is therefore called \emph{posets of type $\mathbb{A}$}. We then give a Geometric Realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type $\mathbb{A}$ as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the Realization of the cluster category of type $\mathbb{A}$ as the category of all diagonals by Caldero, Chapoton and the first author. We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.
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A Geometric Realization of socle-projective categories for posets of type A
Journal of Pure and Applied Algebra, 1Co-Authors: Ralf Schiffler, Robinson-julian SernaAbstract:Abstract This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type A with certain additional arrows. This class of posets is therefore called posets of type A . We then give a Geometric Realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type A as a combinatorial category of certain diagonals of a regular polygon. This construction is inspired by the Realization of the cluster category of type A as the category of all diagonals by Caldero, Chapoton and the first author [10] . We also study the subalgebra of the cluster algebra generated by those cluster variables that correspond to the socle-projectives under the above construction. We give a sufficient condition for when this subalgebra is equal to the whole cluster algebra.
Lucie Jacquetmalo - One of the best experts on this subject based on the ideXlab platform.
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a Geometric Realization of the m cluster categories of type tilde d_n
arXiv: Representation Theory, 2017Co-Authors: Lucie JacquetmaloAbstract:We show that a subcategory of the $m$-cluster category of type $\tilde{D_n}$ is isomorphic to a category consisting of arcs in an $(n-2)m$-gon with two central $(m-1)$-gons inside of it. We show that the mutation of colored quivers and $m$-cluster-tilting objects is compatible with the flip of an $(m+2)$-angulation. In the final part of this paper, we detail an example of a quiver of type $\tilde{D_7}$.