Quaternionic projective space in complex Grassmannian

Question

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 $\mathbb{G}_2(\mathbb{C}^{2n})$"?

Answer 1

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.