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 $\lvert M\rvert-r(M)$ and that it is connected if and only if $M$ is connected. We compute examples including the derived matroids of uniform matroids, the Vámos matroid and the graphical matroid $M(K_4)$. We formulate conjectures relating our construction to previous definitions of derived matroids.
@article{10_37236_11327,
author = {Ragnar Freij-Hollanti and Relinde Jurrius and Olga Kuznetsova},
title = {Combinatorial derived matroids},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {2},
doi = {10.37236/11327},
zbl = {1511.05030},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11327/}
}
TY - JOUR
AU - Ragnar Freij-Hollanti
AU - Relinde Jurrius
AU - Olga Kuznetsova
TI - Combinatorial derived matroids
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/11327/
DO - 10.37236/11327
ID - 10_37236_11327
ER -
