  We prove a root system uniform, concise combinatorial rule for Schubert
calculus of_minuscule_ and_cominuscule_ flag manifolds G/P (the latter are also
known as "compact Hermitian symmetric spaces"). We connect this geometry to the
poset combinatorics of [Proctor '04], thereby giving a generalization of the
[Sch\"{u}tzenberger `77]_jeu de taquin_ formulation of the
Littlewood-Richardson rule that computes the intersection numbers of
Grassmannian Schubert varieties. Our proof introduces_cominuscule recursions_,
a general technique to relate the numbers for different Lie types. A discussion
about connections of our rule to (geometric) representation theory is also
briefly entertained.
