Augmentations, Fillings, and Clusters

Gao, Honghao; Shen, Linhui; Weng, Daping

doi:10.1007/s00039-024-00673-y

Geom. Funct. Anal.

2024

DOI: 10.1007/s00039-024-00673-y

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Augmentations, Fillings, and Clusters

Honghao Gao,

Linhui Shen,

Daping Weng

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2024

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Cited by 4 publications

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A Lagrangian filling for every cluster seed

Casals,

Gao

2024

Invent. math.

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We show that each cluster seed in the augmentation variety contains an embedded exact Lagrangian filling. This resolves the matter of surjectivity of the map from Lagrangian fillings to cluster seeds. The main new technique to produce these Lagrangian fillings is the construction and study of a quiver with potential associated to curve configurations. We prove that its deformation space is trivial and show how to use it to manipulate Lagrangian fillings with $\mathbb{L}$ L -compressing systems via Lagrangian disk surgeries.

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A Lagrangian filling for every cluster seed

Casals,

Gao

2024

Invent. math.

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Cluster structures on braid varieties

Casals,

Gorsky,

Gorsky

et al. 2024

J. Amer. Math. Soc.

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We show the existence of cluster A \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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Configuration Poisson Groupoids of Flags

Mouquin

2022

International Mathematics Research Notices

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Let $G$ be a connected complex semi-simple Lie group and ${\mathcal {B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${{\mathcal {B}}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in ${\mathcal {B}}^n$. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

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Augmentations, Fillings, and Clusters

Cited by 4 publications

References 52 publications

A Lagrangian filling for every cluster seed

A Lagrangian filling for every cluster seed

Cluster structures on braid varieties

Configuration Poisson Groupoids of Flags

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