Schubert cells in real and complex Grassmannians

Fix the coordinate flag $0\subset {\mathbb{F}}^1\subset \cdots \subset {\mathbb{F}}^N$. A Schubert symbol for ${\mathrm{Gr}}_n({\mathbb{F}}^N)$ is a strictly increasing sequence $a=(a_1<\cdots <a_n)$ with $1\le a_i\le N$. Its Schubert cell is

$e(a)=\{W:\dim (W\cap {\mathbb{F}}^{a_i})=i,\dim (W\cap {\mathbb{F}}^{a_i-1})=i-1\text{ for every }i\}\mathrm{.}$

Every $W\in e(a)$ has a unique pivot-normalized basis

$v_i=e_{a_i}+{\sum }_{\begin{array}{c}j<a_i\\ j\mathrm{\notin }\{a_1,\dots ,a_{i-1}\}\end{array}}{x}_{ij}{e}_j,$

after using earlier pivot rows to clear their columns. The free coordinates number

$d(a)={\sum }_{i=1}^n(a_i-i),$

so $e(a)\cong {\mathbb{F}}^{d(a)}$. Its real dimension is $d(a)$ for $\mathbb{F}=\mathbb{R}$ and $2d(a)$ for $\mathbb{F}=\mathbb{C}$. Equivalently, the integers $a_i-i$, read in reverse order, form a partition fitting the $n$ by $(N-n)$ rectangle. These spaces use the Grassmannian topology fixed in Stiefel spaces, Grassmannians, and tautological bundles. For $n=0$, the unique symbol is the empty sequence, the displayed sum is the empty sum $0$, and its Schubert cell is the one-point Grassmannian.