A characterization of positroids, with applications to amalgams and excluded minors
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916347573698560 |
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| author | Bonin, Joseph E. |
| author_facet | Bonin, Joseph E. |
| contents | A matroid of rank $r$ on $n$ elements is a positroid if it has a representation by an $r$ by $n$ matrix over $\mathbb{R}$, each $r$ by $r$ submatrix of which has nonnegative determinant. Earlier characterizations of connected positroids and results about direct sums of positroids involve connected flats and non-crossing partitions. We prove another characterization of positroids of a similar flavor and give some applications of the characterization. We show that if $M$ and $N$ are positroids and $E(M)\cap E(N)$ is an independent set and a set of clones in both $M$ and $N$, then the free amalgam of $M$ and $N$ is a positroid, and we prove a second result of that type. Also, we identify several multi-parameter infinite families of excluded minors for the class of positroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06694 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A characterization of positroids, with applications to amalgams and excluded minors Bonin, Joseph E. Combinatorics A matroid of rank $r$ on $n$ elements is a positroid if it has a representation by an $r$ by $n$ matrix over $\mathbb{R}$, each $r$ by $r$ submatrix of which has nonnegative determinant. Earlier characterizations of connected positroids and results about direct sums of positroids involve connected flats and non-crossing partitions. We prove another characterization of positroids of a similar flavor and give some applications of the characterization. We show that if $M$ and $N$ are positroids and $E(M)\cap E(N)$ is an independent set and a set of clones in both $M$ and $N$, then the free amalgam of $M$ and $N$ is a positroid, and we prove a second result of that type. Also, we identify several multi-parameter infinite families of excluded minors for the class of positroids. |
| title | A characterization of positroids, with applications to amalgams and excluded minors |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.06694 |