Cluster Structures on Double Bott–Samelson Cells

Abstract

Let $\mathsf {C}$ 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.

On exchange matrices from string diagrams

“…In the case of i = ∅ and j corresponding to a reduced word of an element of the Weyl group of the Cartan matrix A, the quiver Q B corresponds to the principal part of the ice quiver constructed in [BIRS09,GLS11] (see Example 3.9). • Generally, the matrix −B T corresponds to the principal part of the matrix constructed in [SW21, Section 3], because we slightly changed the construction of the exchange matrix of a string diagram in [SW21] so that it fits well with the matrices or quivers appearing in [BFZ05,GLS11]. The following is our main result in this paper.…”

“…Cluster algebras A were invented by Fomin and Zelevinsky [FZ02] as a combinatorial approach to the dual canonical bases of quantized enveloping algebras [Lus90,Lus91,Kas90]. Such algebras often arise as the coordinate rings of spaces, such as double Bruhat cells [BFZ05], unipotent cells [GLS11], double Bott-Samelson cells [SW21] and arise as the Grothendieck rings of certain monoidal subcategories of the representations of quantum affine algebras [HL10,HL13].…”

“…Then we can use the string diagram D(S i j (A), T ) to define a skew-symmetrizable matrix B, which is called the exchange matrix of T . The matrix B is closely related with the matrices (or quivers, in skew-symmetric case) appearing in [BFZ05,BIRS09,GLS11,SW21]. More precisely:…”

$\mathscr{A}=\mathscr{U}$ for cluster algebras from moduli spaces of $G$-local systems

“…Our Theorem 1 can be viewed as a generalization of the result of [SW21] from disks to surfaces. • In the other direction, Berenstein, Fomin and Zelevinsky [BFZ05] prove…”

“…We leave it for a future project to achieve a quantum analog of Theorem 1. • Shen and Weng [SW21] prove A = U for cluster algebras associated with double Bott-Samelson cells. Examples of double Bott-Samelson cells include all the double Bruhat cells and the augmentation varieties associated with positive-braid Legendrian links.…”