The Projectivization Matroid of a $q$-Matroid

TL;DR

This paper explores the relationship between $q$-matroids and their associated projectivization matroids, establishing new theoretical connections, recursive formulas for characteristic polynomials, and a $q$-analogue of the critical theorem.

Contribution

It introduces the projectivization map as a functor from $q$-matroids to matroids, linking their properties and characteristic polynomials, and proves a $q$-analogue of the critical theorem.

Findings

The projectivization map is a functor between categories of $q$-matroids and matroids.

The characteristic polynomial of a $q$-matroid equals that of its projectivization matroid.

A recursive formula for the characteristic polynomial of $q$-matroids is established.

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, which allows to prove new results about maps of $q$-matroids. We furthermore show the characteristic polynomial of a $q$-matroid is equal to that of the projectivization matroid. We use this relation 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.