Reference request: Grassmannian and Plucker coordinates in type B, C, D

Question

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.

Answer 1

What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P \hookrightarrow \mathbb P(V_\omega)$ where $V_\omega$ is a fundamental representation (the analogue of the Plücker coordinates). The two coordinate rings, of $\mathbb P(V_\omega)$ and $G/P$, are $Sym(V_\omega^*)$ and $\oplus_n V_{n\omega}^*$ respectively. The kernel of the ring map $Sym(V_\omega^*) \twoheadrightarrow \oplus_n V_{n\omega}^*$ is generated in degree $2$ by Ramanathan's theorem (whose proof you can read in the unique book by Brion + Kumar), i.e. the analogue of the Plücker relations is the complement to $V_{2\omega}^*$ inside $Sym^2(V_\omega^*)$. I don't know enough about that representation theory in the specific $BCD$ cases to tell you more.

Answer 2

In type $B$ and $D$ these are orthogonal isotropic Grassmannians $$OGr(k,n) \subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate quadratic form.

In type $C$ these are symplectic isotropic Grassmannians $$SGr(k,n) \subset Gr(k,n),$$ that parameterize isotropic $k$-dimensional subspaces in a vector space of dimension $n$ endowed with a non-degenerate symplectic form.

Answer 3

It might also be worth mentioning that the Type A Grassmannians are minuscule varieties, meaning they are $G/P$ for a maximal parabolic $P$ corresponding to a minuscule node of the Dynkin diagram. Minuscule (and also cominuscule) varieties tend to behave a bit better than arbitrary homogeneous spaces $G/P$. At least, their combinatorics can be described quite explicitly, like with the Grassmannian: see e.g. https://arxiv.org/abs/math/0608276 or https://arxiv.org/abs/1306.5419. Note that outside of Type A not so many of the nodes of a Dynkin diagram are minuscule.

Answer 4

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".