Recursive structures in the cohomology of flag varieties

Let G be a semisimple complex algebraic group and P be a parabolic subgroup of G and consider the flag variety G/P. The ring H*(G/P) has interesting combinatorial structures with respect to the additive basis of Schubert classes. For example, if G = SL(n) and P is a maximal parabolic, then G/P is a Grassmannian and the structure constants of H*(G/P) with respect to the Schubert classes are governed by Schur polynomials and the Littlewood-Richardson rule. We consider the flag varieties associated to the groups G = SL(n) and Sp(2n) and take P to be any parabolic subgroup (not necessarily maximal). We find that H*(G/P) exhibits Horn recursion on a certain deformation of the cup product. Horn recursion is a term used to describe when non-vanishing products of Schubert classes in H*(G/P) are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of smaller flag varieties. We also find that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written as a product of structure constants coming from induced maximal flag varieties