Computing positroid cells in the Grassmannian of lines, their boundaries and their intersections
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| Format: | Preprint |
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2022
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| _version_ | 1866914027965251584 |
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| author | Agarwala, Susama Mohammadi, Fatemeh Zaffalon, Francesca |
| author_facet | Agarwala, Susama Mohammadi, Fatemeh Zaffalon, Francesca |
| contents | Positroids are families of matroids introduced by Postnikov in the study of non-negative Grassmannians. In particular, positroids enumerate a CW decomposition of the totally non-negative Grassmannian. Furthermore, Postnikov has identified several families of combinatorial objects in bijections with positroids. We will provide yet another characterization of positroids for Gr$_{\geq 0}(2,n)$, the Grassmannians of lines, in terms of certain graphs. We use this characterization to compute the dimension and the boundary of positroid cells. This also leads to a combinatorial description of the intersection of positroid cells, that is easily computable. Our techniques rely on determining different ways to enlarge a given collection of subsets of $\{1,\ldots,n\}$ to represent the dependent sets of a positroid, that is the dependencies among the columns of a matrix with non-negative maximal minors. Furthermore, we provide an algorithm to compute all the maximal positroids contained in a set. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_14001 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Computing positroid cells in the Grassmannian of lines, their boundaries and their intersections Agarwala, Susama Mohammadi, Fatemeh Zaffalon, Francesca Combinatorics Rings and Algebras Positroids are families of matroids introduced by Postnikov in the study of non-negative Grassmannians. In particular, positroids enumerate a CW decomposition of the totally non-negative Grassmannian. Furthermore, Postnikov has identified several families of combinatorial objects in bijections with positroids. We will provide yet another characterization of positroids for Gr$_{\geq 0}(2,n)$, the Grassmannians of lines, in terms of certain graphs. We use this characterization to compute the dimension and the boundary of positroid cells. This also leads to a combinatorial description of the intersection of positroid cells, that is easily computable. Our techniques rely on determining different ways to enlarge a given collection of subsets of $\{1,\ldots,n\}$ to represent the dependent sets of a positroid, that is the dependencies among the columns of a matrix with non-negative maximal minors. Furthermore, we provide an algorithm to compute all the maximal positroids contained in a set. |
| title | Computing positroid cells in the Grassmannian of lines, their boundaries and their intersections |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2206.14001 |