2019
DOI: 10.48550/arxiv.1904.07992
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Cluster Structures on Double Bott-Samelson Cells

Abstract: Let C be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to C. We prove that the decorated double Bott-Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras.We explicitly describe the Donaldson-Thomas transformations on double Bott-Samelson cells and prove that they are cluster transformations. As an application… Show more

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Cited by 10 publications
(26 citation statements)
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“…This is the way we recognize the variety structures on these spaces and perform explicit computations in the subsequent sections. See [SW19] for a description of the variety structure of Conf k P G for k ≥ 3 in terms of the double Bott-Samelson varieties. We also introduce certain functions on these spaces, which will be the local building blocks for Wilson line functions.…”
Section: The Configuration Spacementioning
confidence: 99%
“…This is the way we recognize the variety structures on these spaces and perform explicit computations in the subsequent sections. See [SW19] for a description of the variety structure of Conf k P G for k ≥ 3 in terms of the double Bott-Samelson varieties. We also introduce certain functions on these spaces, which will be the local building blocks for Wilson line functions.…”
Section: The Configuration Spacementioning
confidence: 99%
“…where Conf e w n−2 0 (A ad ) is a double Bott-Samelson variety in [SW19]. In this case, the proof goes through the same steps as in the proof of Theorem 3.44 of [SW19].…”
Section: Proofsmentioning
confidence: 86%
“…Note that O(Σ) is a Poisson algebra. A quasi-cluster transformation of O(Σ) is a Poisson automorphism of O(Σ) that can be obtained by a sequence of mutations followed by a seed isomorphism, e.g., see [SW19,Def.A.29]. Many cluster Poisson algebras contain natural linear bases that are invariant under quasi-cluster transformations.…”
Section: Cluster Poisson Algebrasmentioning
confidence: 99%
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