The Projectivization Matroid of a $q$-Matroid
Abstract
In this paper, we investigate the relation between a -matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the -matroid and considering the projective space as the groundset of the associated matroid on which is defined a rank function compatible with that of the -matroid. We show that the projectivization map is a functor from categories of -matroids to categories of matroids, which allows to prove new results about maps of -matroids. We furthermore show the characteristic polynomial of a -matroid is equal to that of the projectivization matroid. We use this relation to establish a recursive formula for the characteristic polynomial of a -matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a -analogue of the critical theorem in terms of -linear rank metric codes and -matroids.
Keywords
Cite
@article{arxiv.2204.01232,
title = {The Projectivization Matroid of a $q$-Matroid},
author = {Benjamin Jany},
journal= {arXiv preprint arXiv:2204.01232},
year = {2022}
}