# What's the minimal embedding of orthogonal Grassmannian?

Suppose $X$ is the orthogonal Grassmanian. We know the Plücker embedding does not span the whole background $\mathbb{CP}^n,$ just a subspace $\mathbb{CP}^m.$

My question: is there an expression of the isometric embedding of $X$ into $\mathbb{CP}^m$?

By expression, I mean a way to write down explicitly the coordinates such as Plücker coordinates for the Grassmannian?

• I posted a similar question, but in the Lagrangian context. I'm sure that minor modifications will allow you to fit the answer(s) I got to the orthogonal context you're interested in. See: mathoverflow.net/questions/209058/… Sep 30 '15 at 10:13
• By the way, the analogous answer for the embedding in the case of the Lagrangian Grassmannian is this: take the Plucker embedding applied to a symmetric matrix and delete all redundant minors due to the symmetry. The target is now $\bigwedge^n \mathbb{C}^{2n} / \omega\wedge \bigwedge^{n-2} \mathbb{C}^{2n}$, the fundamental representation of the symplectic group with the appropriate highest weight. May 24 '17 at 5:14

## 2 Answers

The image under the Plücker embedding of the isotropic Grassmannian of $r$-planes in $\mathbb{C}^k$ (of course, $r\leq k/2$) spans the subspace generated under $SO(k)$ acting on the highest weight vector $v_1\wedge \cdots \wedge v_r$. This is the entirety of $\bigwedge^{r}\mathbb{C}^k$ with the sole exception of when $k=2r$. In that case, I believe it spans a half dimensional subspace, but to be honest, it is late, and I've forgotten the details.

• I also believe the case $k=2r$ is an exceptional one. In fact, I asked the very same question posted above, but in the skew-symmetric (i.e., symplectic/Lagrangian) context, and the answer turned out to be a little bit trickier than the easy "half dimensional subspace" guess. See: mathoverflow.net/questions/209058/… Sep 30 '15 at 10:10

I stumbled over here looking for something else. I think the short answer is to construct the orthogonal Grassmannian of isotropic n-planes in an 2n-dimensional space, take a list of all the principal pfaffians of a skew-symmetric n by n matrix. Since odd-pfaffians automatically vanish, the construction is slightly different in the even and odd cases.

In general there is a construction of the minimal rational embedding of each compact hermitian symmetric space. In these cases the variety is the unique closed orbit of a highest weight vector and the minimal rational embedding is into the span of the orbit of the highest weight vector. For a complete picture see Landsberg and Manivel's work https://arxiv.org/pdf/math/9902102.pdf