Cox rings of projectivized toric vector bundles and toric flag Bundles

Work of Gonz\'alez, Hering, Payne, and S\"uss shows that it is possible to find both examples and non-examples of Mori dream spaces among projectivized toric vector bundles. This result, and the combinatorial nature of the data of projectivized toric vector bundles make them an ideal test class for the question: what makes a variety a Mori dream space? In the present paper we consider this question with respect to natural algebraic operations on vector bundles. Suppose $\mathcal{E}$ is a toric vector bundle such that the projectivization $\mathbb{P}\mathcal{E}$ is a Mori dream space, then when are the direct sum bundles $\mathbb{P}(\mathcal{E} \oplus \mathcal{E})$, $\mathbb{P}(\mathcal{E} \oplus \mathcal{E} \oplus \mathcal{E})\ldots$ also Mori dream spaces? We give an answer to this question utilizing a relationship with the associated full flag bundle $\mathcal{FL}(\mathcal{E})$. We describe several classes of examples, and we compute a presentation for the Cox ring of the full flag bundle for the tangent bundle of projective space.