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 ∆ A r .When C is of type 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 F q -points we obtain rational functions which are Legendrian link invariants. Contents

Wilson lines and their Laurent positivity

“…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.…”

Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems

“…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].…”

“…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.…”

“…Following Theorem 2.30 in [SW19], the space P G,Dn is a smooth affine variety. The cyclic group Z/n acts on P G,Dn by rotation.…”

“…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]. In fact, as we will see soon, it suffices to first prove the cases when n = 3, 4, which correspond to the double Bruhat cells G e,w 0 and G w 0 ,w 0 ; the theorem follows easily from Theorem 2.10 of [BFZ05].…”