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Braid variety cluster structures, I: 3D plabic graphs

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arxiv 2210.04778 v3 pith:CPAIVM6J submitted 2022-10-10 math.CO math.AGmath.RAmath.RT

Pavel Galashin , Thomas Lam , Melissa Sherman-Bennett , David Speyer This is my paper · ORCID
classification math.COmath.AGmath.RAmath.RT
keywords clusterstructuresvarietiesbraidestablishgraphsopenplabic
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abstract

We introduce $3$-dimensional generalizations of Postnikov's plabic graphs and use them to establish cluster structures for type $A$ braid varieties. Our results include known cluster structures on open positroid varieties and double Bruhat cells, and establish new cluster structures for type $A$ open Richardson varieties.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Decompositions of augmentation varieties via weaves and rulings

    math.SG 2025-08 accept novelty 7.0 of 10

    For positive braids with full Demazure product, the ruling, weave, Deodhar, and sheaf decompositions of the associated variety coincide, and cluster variables can be computed from Morse complex sequences.

  2. Upper cluster structure on Kac--Moody Richardson varieties

    math.RT 2025-06 conditional novelty 7.0 of 10

    Open Richardson varieties in symmetrizable Kac-Moody flag varieties, including twisted-product cases, are shown to carry upper cluster algebra coordinate rings.

  3. Cluster tori over $\mathbb{F}_2$, hexagonal moves on triangulations, and minimal coverings of cluster manifolds

    math.CO 2025-09 conditional novelty 6.0 of 10

    In type A, two seeds determine the same point over F2 exactly when their triangulations are connected by hexagonal moves, and minimal seed sets can cover cluster manifolds over any field, although F2-only coverings al...

  4. Splicing braid varieties

    math.AG 2025-05 conditional novelty 6.0 of 10

    Open sets in braid varieties defined by transversality to a coordinate flag are isomorphic to products of two simpler braid varieties, and in the double Bott-Samelson case this splicing respects cluster structures.

  5. Cluster automorphism group of braid varieties

    math.CO 2025-05 conditional novelty 5.0 of 10

    An explicit inverse matrix A_{u,beta} is constructed for braid varieties; its frozen columns generate the cluster automorphism group action, and the matrix is proven invertible with determinant plus or minus one.

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