TITLE: Plucker relations in orthogonal Grassmannian
QUESTION [5 upvotes]: Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $\varpi_3$. Since $\varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known in literature ? What are the Plücker type relations (quadratic) in this case ? Are the relations same as they are in the Grassmannian $G(3,7)$ ? I have a feeling that the quotient is $\mathbb P^2$.

REPLY [12 votes]: If $G = SO(7)$ and $P$ corresponds to $\varpi_3$, then
$$
G/P \cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).
The isomorphism can be sen as a combination of a general isomorphism
$$
SO(2n-1)/P_{\varpi_{n-1}} \cong SO(2n)/P_{\varpi_{n}} 
$$
and the triality isomorphism 
$$
SO(8)/P_{\varpi_{4}} \cong SO(8)/P_{\varpi_{1}} \cong Q^6.
$$