Poset polytopes and pipe dreams: toric degenerations and beyond

We demonstrate how pipe dreams can be applied to the theory of poset polytopes to produce toric degenerations of flag varieties. Specifically, we present such constructions for marked chain-order polytopes of Dynkin types A and C. These toric degenerations also give rise to further algebraic and geometric objects such as PBW-monomial bases and Newton–Okounkov bodies. We discuss a construction of the former in the type A case and of the latter in type C.