Cominuscule Schubert calculus and tableau combinatorics

I will discuss 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 work of [Proctor'04] in poset combinatorics, thereby generalizing [Schützenberger'77]'s jeu de taquin formulation of the Littlewood-Richardson rule for computing intersection numbers of Grassmannian Schubert varieties.

The proof uses cominuscule recursions, a technique relating Schubert numbers for different Lie types. I will also discuss cominuscule dual equivalence, a generalization of a concept due to [Haiman'92]. We use this to provide an independent proof of Proctor's results needed in our context, as well as to extend tableau combinatorics such as [Fomin'88]'s growth diagrams and [Schützenberger'63]'s evacuation involution.

This talk is based on math.AG/0608276 and math.CO/0701215.