$\DeclareMathOperator\GL{GL}$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.



Littlewood–Richardson–Type Rule for Cohomology Ring of Grassmannians
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](https://doi.org/10.1016/j.aim.2009.05.008), arXiv:[math/0608276](https://arxiv.org/abs/math/0608276)


for details.


