Cluster structures on braid varieties
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866915006439751680 |
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| author | Casals, Roger Gorsky, Eugene Gorsky, Mikhail Le, Ian Shen, Linhui Simental, José |
| author_facet | Casals, Roger Gorsky, Eugene Gorsky, Mikhail Le, Ian Shen, Linhui Simental, José |
| contents | 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 on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_11607 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cluster structures on braid varieties Casals, Roger Gorsky, Eugene Gorsky, Mikhail Le, Ian Shen, Linhui Simental, José Representation Theory Algebraic Geometry Combinatorics Symplectic Geometry 13F60, 14M15 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 on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties. |
| title | Cluster structures on braid varieties |
| topic | Representation Theory Algebraic Geometry Combinatorics Symplectic Geometry 13F60, 14M15 |
| url | https://arxiv.org/abs/2207.11607 |