I would like to consider the quaternionic projective space $\mathbb{PH}^{n-1}\subset\mathbb{G}\_2(\mathbb{C}^{2n})$ as a subvariety of the Grassmannian of complex 2-planes. 
For a real vector $e\in\mathbb{R}^{4n}$ the inclusion is just given by
$$
\mathbb{H}\cdot e
=
\mathbb{C}\cdot e
\oplus
\mathbb{C}\cdot Je.
$$
One has
$$
\mathbb{PH}^{n-1}
=
\{V\in\mathbb{G}\_2(\mathbb{C}^{2n})\colon JV=V\}.
$$


My question is: "how to compute the fundamental class of
$\mathbb{PH}^{n-1}$
in
$\mathbb{G}\_2(\mathbb{C}^{2n})$"?



Quaternionic projective space in complex Grassmannian
Here is how I understand David Treumann indications.


* We take a flag $F$ of $\mathbb{C}^{2n}$ such that $JF$ is the opposite flag.
* The complex codimension of $\mathbb{HP}^{n-1}$ in $\mathbb{G}\_2(\mathbb{C}^{2n})$ being $2(n-1)$ (half of the dimension of the Grassmannian), we consider middle dimensional Schubert varieties $X\_\lambda$, associated to partitions $\lambda$ of length $2(n-1)$ with at most two parts, i.e. of the form $(2n-1-i,i-1)$ for $i=1,\dots,n$.
* These Schubert varieties are defined by the incidence conditions
$$\dim(V\cap F\_i)\geq 1\text{ and }\dim(V\cap F\_{2n+1-i})\geq 2,$$
that is
$$\dim(V\cap F\_i)\geq 1\text{ and }\dim(V\cap JF\_{i-1})=0.$$
So $\mathbb{HP}^{n-1}$ meets each of these middle dimensional Schubert varieties transversely in one point.
* Thus in the Schubert basis $[\mathbb{HP}^{n-1}]=\sum\_{i=1}^{n}[X\_{2n-1-i,i-1}]$. By Giambelli formula this is
$$[\mathbb{HP}^{n-1}]=\sum\_{i=1}^{n}s\_{2n-1-i,i-1}(U)=s\_{n-1}(U)s\_{n-1}(U),$$
where (U) is the universal subbundle on the Grassmann bundle.


