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, we complete the proof of the Fock-Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson-Thomas transformations on a family of double Bott-Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov's periodicity conjecture in the cases of $\Delta \square {\mathrm{A}}_r$. When $C$ is of type $\mathrm{A}$, the double Bott-Samelson cells are isomorphic to Shende-Treumann-Zaslow's moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their ${\mathbb{F}}_q$-points we obtain rational functions which are Legendrian link invariants.

109 pages

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The quantum dilogarithm and representations quantum cluster varieties
• Quantum geometry of moduli spaces of local systems and representation theory
• Cluster algebras IV: Coefficients
• Cyclic Sieving and Cluster Duality of Grassmannian
• Cluster algebra structures on Poisson nilpotent algebras

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