String cone and superpotential combinatorics for flag and Schubert varieties in type A
Lara Bossinger,Ghislain Fourier +1 more
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In this paper, the authors studied the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties, and they proved that the two cones are unimodularly equivalent.About:
This article is published in Journal of Combinatorial Theory, Series A.The article was published on 2019-10-01 and is currently open access. It has received 11 citations till now. The article focuses on the topics: Flag (geometry) & Superpotential.read more
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Families of Gröbner degenerations, Grassmannians and universal cluster algebras
TL;DR: In this paper, a generalization of the classical one-parameter Grobner degeneration associated to a weight has been proposed, which is the pull-back of a toric family defined by a Rees algebra along the universal torsor.
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Toric degenerations of cluster varieties and cluster duality
TL;DR: In this paper, the notion of a cluster duality was introduced and used to link the Batyrev-Borisov duality to the Toric Fanos duality in the context of mirror symmetry.
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Polyhedral parametrizations of canonical bases & cluster duality
TL;DR: In this article, the relation of Berenstein-Kazhdan's decoration function and Gross-Hacking-Keel-Kontsevich's potential on the open double Bruhat cell in the base affine space G/N of a simple, simply connected, simply laced algebraic group G was established.
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Combinatorics of canonical bases revisited: Type A
TL;DR: In this article, the combinatorics of several parametrizations of canonical bases of Lie algebras of type A were studied using geometric objects called rhombic tilings and a crossing formula for the action of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element.
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Toric degenerations of cluster varieties and cluster duality
TL;DR: In this article, the authors introduce the notion of a $Y$-pattern with coefficients and its geometric counterpart: a cluster $\mathcal{X}$-variety with coefficients.
References
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Journal ArticleDOI
Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.
Book ChapterDOI
polymake: a Framework for Analyzing Convex Polytopes
Ewgenij Gawrilow,Michael Joswig +1 more
TL;DR: An overview of the functionality as well as of the structure of the polymake tool is given, seen as a first approximation to a polymake handbook.
Journal Article
Finite-dimensional representations of the group of unimodular matrices
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Cluster algebras III: Upper bounds and double Bruhat cells
TL;DR: In this paper, an upper cluster algebra is defined as an intersection of Laurent polynomial rings, and it is shown that under an assumption of ''acyclicity'' a cluster algebra coincides with its upper counterpart and is finitely generated; in this case, its defining ideal and construct a standard monomial basis.
Book
Introduction to Tropical Geometry
Diane Maclagan,Bernd Sturmfels +1 more
TL;DR: Tropical islands Building blocks Tropical varieties Tropical rain forest Tropical garden Toric connections Bibliography Index Bibliography as mentioned in this paper, Section 5.1.1] and Bibliography 2.2.
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Standard monomial theory and toric degenerations of Schubert varieties from matching field tableaux
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