Littlewood–Richardson–Type Rule for Cohomology Ring of Grassmannians

Question

$\DeclareMathOperator\GL{GL}$The ordinary Grassmannian of k-planes in n-space is a coset space for $\GL_n$. It is $\GL_n$ mod a maximal parabolic. Here there is a nice basis given by Schubert varieties, which can be indexed by Young diagrams that fit in a $(k)\times(n-k)$ box. The structure constants for the cup product are then given by Littlewood–Richardson numbers.

My question: is there a similarly nice picture for Grassmannians of arbitrary simple groups? Here the ordinary Grassmannian is replaced by $G/P$ where $G$ is a simple group and $P$ is a maximal parabolic. There are still Schubert varieties in this case, but I don't know how to say anything about the cup product.

Answer 1

As yet, such a nice rule has only been formulated in the case that $G/P$ is minuscule or co-minuscule. See

• Hugh Thomas, Alexander Yong, A combinatorial rule for (co)minuscule Schubert calculus, Adv. Math. 222 (2009), no. 2, 596–620, doi:10.1016/j.aim.2009.05.008, arXiv:math/0608276

for details.