Ternary and quaternary positroids
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913426217893888 |
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| author | Quail, Jeremy |
| author_facet | Quail, Jeremy |
| contents | A positroid is an ordered matroid realizable by a real matrix with all nonnegative maximal minors. Postnikov gave a map from ordered matroids to Grassmann necklaces, for which there is a unique positroid in each fiber of the map. Here, we give forbidden minor characterizations of ternary and quaternary positroids. We show that a positroid is ternary if and only if it is near-regular, and that all ternary positroids are formed by direct sums and $2$-sums of binary positroids and positroid ordered whirls. We prove that a positroid is quaternary if and only if it is $U^2_6, U^4_6,$ and $P_6$-free. Under the map from ordered matroids to Grassmann necklaces, we fully characterize the fibers of ternary positroids, referred to as their positroid envelope classes; in particular, the envelope class of a positroid ordered whirl of rank-$r$ contains exactly four matroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06956 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ternary and quaternary positroids Quail, Jeremy Combinatorics 05B35 A positroid is an ordered matroid realizable by a real matrix with all nonnegative maximal minors. Postnikov gave a map from ordered matroids to Grassmann necklaces, for which there is a unique positroid in each fiber of the map. Here, we give forbidden minor characterizations of ternary and quaternary positroids. We show that a positroid is ternary if and only if it is near-regular, and that all ternary positroids are formed by direct sums and $2$-sums of binary positroids and positroid ordered whirls. We prove that a positroid is quaternary if and only if it is $U^2_6, U^4_6,$ and $P_6$-free. Under the map from ordered matroids to Grassmann necklaces, we fully characterize the fibers of ternary positroids, referred to as their positroid envelope classes; in particular, the envelope class of a positroid ordered whirl of rank-$r$ contains exactly four matroids. |
| title | Ternary and quaternary positroids |
| topic | Combinatorics 05B35 |
| url | https://arxiv.org/abs/2403.06956 |