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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!

Matt Samuel
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ai.jennetta
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  • I think there might be a typo, as I'm pretty sure $\mathcal{B}$ should be "the set of all full flags", not "the set of all full flag varieties". So an element is a full flag as they described. – Elizabeth S. Q. Goodman Apr 07 '17 at 07:49
  • For the question of how $GL_n(F)$ acts on $\mathcal{B}$, we need to know at minimum: (1) is there a sensible way to make any invertible $n\times n$ matrix $M$ transform a full flag? I would say yes: multiply $M$ by every vector in the subspaces. (2) Is the result always a full flag? (It needs to be, to have a group action.) – Elizabeth S. Q. Goodman Apr 07 '17 at 07:53
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    OK this part is beyond the level of hint you asked for. But it's also helpful when showing an action exists, to describe also: (3) What properties are preserved under this action? (4) Why did they pick $GL_n(F)$ instead of $M_n(F)$, i.e. why invertible matrices instead of all $n\times n$ matrices? (5) Is the action transitive? What about free? etc. – Elizabeth S. Q. Goodman Apr 07 '17 at 07:55
  • @ElizabethS.Q.Goodman you're right, I fixed it. – ai.jennetta Apr 07 '17 at 08:11
  • Does that help though or are you still stuck? – Elizabeth S. Q. Goodman Apr 07 '17 at 08:20
  • @ElizabethS.Q.Goodman between your hints and N.H. I understand this problem now, thank you both so much! – ai.jennetta Apr 07 '17 at 12:51

1 Answers1

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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 $F^4$.