6 a Combinatorial Rule for ( Co ) Minuscule Schubert Calculus Hugh

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 [Proc-tor '04], thereby giving a generalization of the [Schützenberger '77] jeu de taquin formulation of the Littlewood-Richardson rule that computes the intersection numbers of Grassman-nian 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.