Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas)
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929194781376512 |
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| author | Casals, Roger Le, Ian Sherman-Bennett, Melissa Weng, Daping |
| author_facet | Casals, Roger Le, Ian Sherman-Bennett, Melissa Weng, Daping |
| contents | First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_06184 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas) Casals, Roger Le, Ian Sherman-Bennett, Melissa Weng, Daping Combinatorics Representation Theory Symplectic Geometry 13F60, 14M15, 53D12 First, this article develops the theory of weaves and their cluster structures for the affine cones of positroid varieties. In particular, we explain how to construct a weave from a reduced plabic graph, show it is Demazure, compare their associated cluster structures, and prove that the conjugate surface of the graph is Hamiltonian isotopic to the Lagrangian filling associated to the weave. The T-duality map for plabic graphs has a surprising key role in the construction of these weaves. Second, we use the above established bridge between weaves and reduced plabic graphs to show that the Muller-Speyer twist map on positroid varieties is the Donaldson-Thomas transformation. This latter statement implies that the Muller-Speyer twist is a quasi-cluster automorphism. An additional corollary of our results is that target labeled seeds and the source labeled seeds are related by a quasi-cluster transformation. |
| title | Demazure weaves for reduced plabic graphs (with a proof that Muller-Speyer twist is Donaldson-Thomas) |
| topic | Combinatorics Representation Theory Symplectic Geometry 13F60, 14M15, 53D12 |
| url | https://arxiv.org/abs/2308.06184 |