The Projectivization Matroid of a $q$-Matroid

Abstract

In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-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 $q$-matroid. We show that the projectivization map is a functor from categories of $q$-matroids to categories of matroids. This relation is used to prove new results about maps of $q$-matroids. Furthermore, we show the characteristic polynomial of a $q$-matroid is equal to that of the projectivization matroid, which we use to establish a recursive formula for the characteristic polynomial of a $q$-matroid in terms of the characteristic polynomial of its minors. Finally we use the projectivization matroid to prove a $q$-analogue of the critical theorem in terms of $\mathbb{F}_{q^m}$-linear rank metric codes and $q$-matroids