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We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig’s coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc [Adv. Math. 300 (2016), pp. 190–228] on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties. 

abstract: In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting decompositions of braid varieties and their quotients. It is shown that the maximal charts of these decompositions are exponential Darboux charts for the holomorphic symplectic structures, and we relate these charts to exact Lagrangian fillings of Legendrian links. 

Abstract The double Dyck path algebra$$\mathbb {A}_{q,t}$$ $A_{q,t}$and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$. We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces. 

Abstract These notes cover the lectures of the first named author at 2021 IHES Summer School on “Enumerative Geometry, Physics and Representation Theory” with additional details and references. They cover the definition of Khovanov‐Rozansky triply graded homology, its basic properties and recent advances, as well as three algebro‐geometric models for link homology: braid varieties, Hilbert schemes of singular curves and affine Springer fibers, and Hilbert schemes of points on the plane.