Abstract
AbstractLet$\mathsf {C}$be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to$\mathsf {C}$. 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$\mathsf {C}$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 that are Legendrian link invariants.
Highlights
We introduce a new family of varieties called double Bott–Samelson cells as a natural generalisation of double Bruhat cells and study their cluster structures
Our generalisation goes in two directions: first, we extend the groups from semisimple types to Kac–Peterson groups, whose double Bruhat cells have been studied by Williams [Wil13]; second, we replace a pair of Weyl group elements (, ) by a pair of positive braids (, ), which we believe is a new construction
We prove the following result on cluster structures of double Bott–Samelson cells
Summary
We construct seeds that define the cluster structures on double Bott–Samelson cells. We call the numbers the cluster Poisson coordinates associated to the seed (string diagram/triangulation) on Conf (Aad). For a triangulation chosen as above and any closed string c in the corresponding string diagram, the once-mutated cluster K2 variable ′ (as an element in Frac O a priori) belongs to O and Conf (Asc) contains the seed torus Spec C A±. A using the K2 cluster associated to the following triangle: To determine the last decorated flag A +1, we need to compute Δ ( + 1, + 1) using Equation (3.41); note that the numerical value of everything else in that equation is already given and the coefficient of Δ ( + 1, + 1) is Δ ( , −1), which is nonzero.
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