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Graphic Positroids and Positroid Envelope Classes
Quail, Jeremy Albertoemilo
Quail, Jeremy Albertoemilo
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Abstract
Matroids are combinatorial objects that generalize vector configurations. We study a subclass of ordered matroids, called positroids, that can be represented by real matrices with all nonnegative maximal minors. We do so by leveraging Grassmann necklaces and decorated permutations, two classes of combinatorial objects that are known to be in bijection with positroids. Both Grassmann necklaces and decorated permutations yield identical partitions of the set of ordered matroids into equivalences classes, called positroid envelope classes. We give an explicit graph construction, using decorated permutations, that shows that every positroid envelope class contains a graphic matroid. We prove that a graphic positroid is the unique matroid contained in its envelope class. We show that every graphic positroid has an oriented graph representable by a signed incidence matrix with all nonnegative maximal minors. A matroid is ternary or quaternary if it can be represented by a matrix with entries over the finite field of three or four elements respectively. We give forbidden minor characterizations of ternary and quaternary positroids. A positroid is ternary if and only if it is near-regular, and all ternary positroids are formed by direct sums and 2-sums of binary positroids and positroid ordered whirls. We fully characterize the ternary positroid envelope classes; in particular, the envelope class of a positroid ordered whirl of rank-r contains exactly four matroids.
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2025-01-01
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Mathematical Sciences