TITLE: Reference request: Grassmannian and Plucker coordinates in type B, C, D
QUESTION [16 upvotes]: Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker relations in these cases? Are there some references about this? Thank you very much.

REPLY [2 votes]: For the orthogonal case (B and D), this was developed in great detail in Elie Cartan's "Theory of Spinors" (Dover Publications Inc, Mineola N.Y., 1981); see also:
Claude Chevalley, "The Algebraic Theory of Spinors and Clifford Algebras" (Springer, 1996, Vol. 2 of collecteed works). Of special importance are the "maximal" isotropic Grassmannians, which are related, in Cartan's approach, to "pure spinors". The analog of the Plucker relations are the "Cartan relations", or "Pfaffian Plucker relations".
Further details and developments may be found in:
J. Harnad and S.Shnider,  "Isotropic geometry and twistors in higher dimensions
I. The generalized Klein correspondence and spinor flags in even dimensions,"  J. Math. Phys. 33, 3191-3208 (1992) and
J. Harnad and S.Shnider, "Isotropic geometry and twistors in higher dimensions II. Odd dimensions, reality conditions and twistor superspaces",
J. Math. Phys. 36 1945-1970 (1995).
For the symplectic case (C), again, the "maximal isotropic" case is of special importance, or "Lagrangian Grassmannians".