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Joel Kamnitzer - One of the best experts on this subject based on the ideXlab platform.
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Mirkovic-Vilonen cycles and Polytopes
Annals of Mathematics, 2010Co-Authors: Joel KamnitzerAbstract:We give an explicit description of the Mirkovie-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV Polytopes. We prove that a Polytope is an MV Polytope if and only if it is a lattice Polytope whose defining hyperplanes are parallel to those of the Weyl Polytopes and whose 2-faces are rank 2 MV Polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV Polytopes.
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Mirkovic-Vilonen cycles and Polytopes
arXiv: Algebraic Geometry, 2005Co-Authors: Joel KamnitzerAbstract:We give an explicit description of the Mirkovic-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV Polytopes. We prove that a Polytope is an MV Polytope if and only if it a lattice Polytope whose defining hyperplanes are parallel to those of the Weyl Polytopes and whose 2-faces are rank 2 MV Polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV Polytopes.
Dido Salazar - One of the best experts on this subject based on the ideXlab platform.
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Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes
Discrete Mathematics and Theoretical Computer Science, 2011Co-Authors: Federico Ardila, Thomas Bliem, Dido SalazarAbstract:Stanley (1986) showed how a finite partially ordered set gives rise to two Polytopes, called the order Polytope and chain Polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of Polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin Polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg Polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin Polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg Polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
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Gelfand-Tsetlin Polytopes and Feigin-Fourier-Littelmann-Vinberg Polytopes as marked poset Polytopes
Journal of Combinatorial Theory Series A, 2011Co-Authors: Federico Ardila, Thomas Bliem, Dido SalazarAbstract:Stanley (1986) showed how a finite partially ordered set gives rise to two Polytopes, called the order Polytope and chain Polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of Polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin Polytopes (1950) and the Feigin-Fourier-Littelmann-Vinberg Polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand-Tsetlin Polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin-Fourier-Littelmann-Vinberg Polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
Federico Ardila - One of the best experts on this subject based on the ideXlab platform.
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Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes
Discrete Mathematics and Theoretical Computer Science, 2011Co-Authors: Federico Ardila, Thomas Bliem, Dido SalazarAbstract:Stanley (1986) showed how a finite partially ordered set gives rise to two Polytopes, called the order Polytope and chain Polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of Polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin Polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg Polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin Polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg Polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
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Gelfand-Tsetlin Polytopes and Feigin-Fourier-Littelmann-Vinberg Polytopes as marked poset Polytopes
Journal of Combinatorial Theory Series A, 2011Co-Authors: Federico Ardila, Thomas Bliem, Dido SalazarAbstract:Stanley (1986) showed how a finite partially ordered set gives rise to two Polytopes, called the order Polytope and chain Polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of Polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin Polytopes (1950) and the Feigin-Fourier-Littelmann-Vinberg Polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand-Tsetlin Polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin-Fourier-Littelmann-Vinberg Polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
Vyacheslav P Grishukhin - One of the best experts on this subject based on the ideXlab platform.
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delaunay and voronoi Polytopes of the root lattice e7 and of the dual lattice e 7
Proceedings of the Steklov Institute of Mathematics, 2011Co-Authors: Vyacheslav P GrishukhinAbstract:We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi Polytopes of the root lattice E7 and the dual lattice E*7. Here three uniform Polytopes related to the Coxeter-Dynkin diagram of the Lie algebra E7 play a special role. These are the Gosset Polytope PGos = 321, which is a Delaunay Polytope, the contact Polytope 231 (both for the lattice E7), and the Voronoi Polytope PV(E*7) = 132 of the dual lattice E*7. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi Polytopes.
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Delaunay and voronoi Polytopes of the root lattice E _7 and of the dual lattice E*_7
Proceedings of the Steklov Institute of Mathematics, 2011Co-Authors: Vyacheslav P GrishukhinAbstract:We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi Polytopes of the root lattice E _7 and the dual lattice E *_7. Here three uniform Polytopes related to the Coxeter-Dynkin diagram of the Lie algebra E _7 play a special role. These are the Gosset Polytope P _Gos = 3_21, which is a Delaunay Polytope, the contact Polytope 2_31 (both for the lattice E _7), and the Voronoi Polytope P _V( E *_7) = 1_32 of the dual lattice E *_7. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi Polytopes.
Thomas Bliem - One of the best experts on this subject based on the ideXlab platform.
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Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes
Discrete Mathematics and Theoretical Computer Science, 2011Co-Authors: Federico Ardila, Thomas Bliem, Dido SalazarAbstract:Stanley (1986) showed how a finite partially ordered set gives rise to two Polytopes, called the order Polytope and chain Polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of Polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin Polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg Polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin Polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg Polytopes corresponding to the symplectic and odd orthogonal Lie algebras.
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Gelfand-Tsetlin Polytopes and Feigin-Fourier-Littelmann-Vinberg Polytopes as marked poset Polytopes
Journal of Combinatorial Theory Series A, 2011Co-Authors: Federico Ardila, Thomas Bliem, Dido SalazarAbstract:Stanley (1986) showed how a finite partially ordered set gives rise to two Polytopes, called the order Polytope and chain Polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of Polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand-Tsetlin Polytopes (1950) and the Feigin-Fourier-Littelmann-Vinberg Polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand-Tsetlin Polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin-Fourier-Littelmann-Vinberg Polytopes corresponding to the symplectic and odd orthogonal Lie algebras.