Finest positroid subdivisions from maximal weakly separated collections
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918207504252928 |
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| author | Koshevoy, Gleb A. Li, Fang Zhang, Lujun |
| author_facet | Koshevoy, Gleb A. Li, Fang Zhang, Lujun |
| contents | We adopt a formal and algebraic approach of Early \cite{E2} to study the positive tropical Grassmannian $\operatorname{Trop}^+ Gr_{k,n}$. Specifically, we deal with positroid subdivision of hypersimplex induced by translated blades from any maximal weakly separated collection. One of our main results gives a necessary and sufficient condition on a maximal weakly separated collection to form a positroid subdivision of a hypersimplex corresponding to a simplicial cone in $\rm Trop^+Gr_{k,n}$. For k = 2 our condition says that any weakly separated collection of two-elements sets gives such a simplicial cone, and all cones are of such a form. We also show that the maximality of any weakly separated collection is preserved under the boundary map, which armatively answers a question by Early in \cite{E1}. Plabic graphs, invented by Postnikov \cite{P}, are of use in proving this result. As a corollary, we get that all those positroid subdivisions are the finest. Thus, the flip of two maximal weakly separatedcollections corresponds to a pair of adjacent maximal cones in positive tropical Grassmannian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finest positroid subdivisions from maximal weakly separated collections Koshevoy, Gleb A. Li, Fang Zhang, Lujun Representation Theory Combinatorics We adopt a formal and algebraic approach of Early \cite{E2} to study the positive tropical Grassmannian $\operatorname{Trop}^+ Gr_{k,n}$. Specifically, we deal with positroid subdivision of hypersimplex induced by translated blades from any maximal weakly separated collection. One of our main results gives a necessary and sufficient condition on a maximal weakly separated collection to form a positroid subdivision of a hypersimplex corresponding to a simplicial cone in $\rm Trop^+Gr_{k,n}$. For k = 2 our condition says that any weakly separated collection of two-elements sets gives such a simplicial cone, and all cones are of such a form. We also show that the maximality of any weakly separated collection is preserved under the boundary map, which armatively answers a question by Early in \cite{E1}. Plabic graphs, invented by Postnikov \cite{P}, are of use in proving this result. As a corollary, we get that all those positroid subdivisions are the finest. Thus, the flip of two maximal weakly separatedcollections corresponds to a pair of adjacent maximal cones in positive tropical Grassmannian. |
| title | Finest positroid subdivisions from maximal weakly separated collections |
| topic | Representation Theory Combinatorics |
| url | https://arxiv.org/abs/2502.05033 |