  We prove that Schubert varieties in potentially different Grassmannians are
isomorphic as varieties if and only if their corresponding Young diagrams are
identical up to a transposition. We also discuss a generalization of this
result to Grassmannian Richardson varieties. In particular, we prove that
Richardson varieties in potentially different Grassmannians are isomorphic as
varieties if their corresponding skew diagrams are semi-isomorphic as posets,
and we conjecture the converse. Here, two posets are said to be semi-isomorphic
if there is a bijection between their sets of connected components such that
the corresponding components are either isomorphic or opposite.
