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arXiv · 1702.00385

Braid group symmetries of Grassmannian cluster algebras

Chris Fraser
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Abstract

We define an action of the extended affine d-strand braid group on the open positroid stratum in the Grassmannian Gr(k,n), for d the greatest common divisor of k and n. The action is by quasi-automorphisms of the cluster structure on the Grassmannian, determining a homomorphism from the extended affine braid group to the cluster modular group. We also define a quasi-isomorphism between the Grassmannian Gr(k,rk) and the Fock-Goncharov configuration space of 2r-tuples of affine flags for SL(k). This identifies the cluster variables, clusters, and cluster modular groups, in these two cluster structures. Fomin and Pylyavskyy proposed a description of the cluster combinatorics for Gr(3,n) in terms of Kuperberg's basis of non-elliptic webs. As our main application, we prove many of their conjectures for Gr(3,9) and give a presentation for its cluster modular group. We establish similar results for Gr(4,8). These results rely on the fact that both of these Grassmannians have finite mutation type.

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Chris Fraser. 2018-08-16. Braid group symmetries of Grassmannian cluster algebras. https://arxiv.org/abs/1702.00385

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