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arXiv · 2206.06881

Combinatorial Derived Matroids

Olga Kuznetsova·Ragnar Freij-Hollanti·Relinde Jurrius
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Abstract

Let $M$ be an arbitrary matroid with circuits $\mathcal{C}(M)$. We propose a definition of a derived matroid $\delta M$ that has as its ground set $\mathcal{C}(M)$. Unlike previous attempts of such a definition, our definition applies to arbitrary matroids, and is completely combinatorial. We prove that the rank of $\delta M$ is bounded from above by $|M|-r(M)$, that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the V\'amos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.

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Olga Kuznetsova, Ragnar Freij-Hollanti, Relinde Jurrius. 2022-06-14. Combinatorial Derived Matroids. https://arxiv.org/abs/2206.06881

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