Group $ GL(V) $ acts naturally on $ \mathscr{F} $, how to get its orbits?

Question

Could you please explain to me the meaning of the marking? Thanks in advance.

Answer 1

This is basics of flag varieties.

$GL(V)$ acts transitively on each set of flags $V_1 \subset \dots \subset V_n$ where the dimensions of the subspaces $V_i$ are fixed (this is basic linear algebra).

Equivalently, we can describe the dimensions of the subspaces $V_i$ by giving $dim V_1$ and the dimensions of the quotient vector spaces $V_{i+1}/V_i$ (which is simply $dim V_{i+1} - dim V_i$.

Answer 2

What it is saying is that if you have two flags $$ f=(V_1\subset\cdots\subset V_n)\quad\text{and}\quad g=(W_1\subset\cdots\subset W_n) $$ in $\cal F$, you can find $g\in{\rm GL}(V)$ such that $$ g(V_i)=W_i\quad\text{for all}\quad i=1,...,n $$ if and only if $$ \dim(V_{i+1}/V_i)=\dim(W_{i+1}/W_i)\quad\text{for all}\quad i=1,...,n. $$