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Cluster structures on braid varieties

https://doi.org/10.1090/jams/1048

Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Le, Ian; Shen, Linhui; Simental, José (April 2025, Journal of the American Mathematical Society)

We show the existence of cluster $A$ -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.
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Free, publicly-accessible full text available April 1, 2026
Algebraic weaves and braid varieties

https://doi.org/10.1353/ajm.2024.a944357

Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Simental, José (December 2024, American Journal of Mathematics)

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.
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Free, publicly-accessible full text available December 1, 2025
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Krattenthaler, Christian; Thibon, Jean-Yves (Ed.)
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