I have the following 2-part question as a homework assignment...
Let $F$ be a field and $n\in \mathbb{Z}_{\geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces \begin{align} \{0\}\subset V_1\subset \cdots\subset V_{n-1}\subset V_n=F^n\end{align} such that $\dim V_i=i$ for all $i\in \{1,2,...,n\}$. The flag variety of $GL_n(F)$ is the set $\mathcal{B}$ of all full flags in $F^n$.
(a) Show that $GL_n(F)$ acts on the flag variety $\mathcal{B}$.
(b) Let $e_i$ be the coordinate basis vectors of $F^n$. Let $e_{*}$ be the full flag given by $V_i=\mathrm{span}\{e_1,...,e_i\}$. Show that $\mathrm{Stab}_{GL_n(F)}(e_*)=B$, where $B$ is the subgroup of upper-triangular matrices.
I feel like once i understand how to tackle part (a), I will understand part (b), but at the moment I really don't understand what this flag variety really is.
$\quad$-What does an element of the flag variety look like?
$\quad$-How would $GL_n(F)$ act on such an element?
These are the things I need some hints on. Thank you!
1) If $V_1 \subset V_2 \subset \dots \subset V_n$ is a flag, and $g \in GL_n(F)$ then $gV_1 \subset gV_2 \subset \dots \subset gV_n$ is a flag. This is the action of $GL_n(F)$ on $\mathcal B$.
2) Let $g \in Stab(e_*)$. This mean that $g V_i \subset V_i$ for all $i$. For $i = 1$, we want $g e_1 \in \text{span} \{e_1\}$ by definition so the first column of $g$ is $\begin{pmatrix} \lambda \\ 0 \\ \dots \\ 0\end{pmatrix}$
Next step is $gV_2 \subset V_2$, i.e $g e_2 \in \text{span}\{e_1, e_2\}$ i.e the second column of $g$ will be $\begin{pmatrix} \mu \\ \gamma \\ 0 \\ \dots \\ 0\end{pmatrix}$. I think you can continue easily this describtion.
3) If you want to be sure you understood correctly, you can try to find the stabilizer of the incomplete flag $\{e_1\} \subset \{e_1, e_2, e_3\}$ in $F^4$.