Cluster structures on braid varieties

Casals, Roger; Gorsky, Eugene; Gorsky, Mikhail; Le, Ian; Shen, Linhui; Simental, José

doi:10.1090/jams/1048

Cluster structures on braid varieties
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by Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, Linhui Shen and José Simental;

J. Amer. Math. Soc.

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

Published electronically: October 1, 2024

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Abstract:

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.

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