Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as that $p$ is odd, $p>k$, etc.) What are the cohomology algebras
$$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$
$$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$
I have obtained
$$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,c_k]/(\bar w_{N-k+1},\bar w_{N-k+2},\cdots,\bar w_N). $$