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arXiv · 2306.06694

A characterization of positroids, with applications to amalgams and excluded minors

Joseph E. Bonin
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Abstract

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.

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Joseph E. Bonin. 2023-06-11. A characterization of positroids, with applications to amalgams and excluded minors. https://doi.org/10.1016/j.ejc.2024.104040

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