Cox rings of projectivized toric vector bundles and toric flag bundles

Work of González, Hering, Payne, and Süß 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 toric vector bundles. Suppose E is a toric vector bundle such that the projectivization PE is a Mori dream space, then when are the direct sum toric vector bundles P(E⊕E), P(E⊕E⊕E)… also Mori dream spaces? We give an answer to this question utilizing a relationship with the associated full flag bundle FL(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.