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$\DeclareMathOperator\GL{GL}$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.

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

for details.

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    $\begingroup$ ...and I was just trying to work exactly this out. Good reference, David. $\endgroup$ Nov 9, 2009 at 18:25
  • $\begingroup$ I'm a little late to point this out, but, in type $A$ all maximal $P$ are cominuscule so one doesn't notice the importance of this condition. $\endgroup$ Oct 13, 2022 at 11:57

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