Algebra and Geometry of Irreducible toric vector bundles of rank $n$ on ${\mathbb{P}}^n$

We construct a presentation for the Cox ring of the projectivization $\mathbb{P}\mathcal{E}$ of any rank $n$ irreducible toric vector bundle on ${\mathbb{P}}^n$. We use this presentation to show that $\mathbb{P}\mathcal{E}$ always satisfies Fujita’s freeness and ampleness conjectures.