$GL_n(F)$ acts on the flag variety
$begingroup$
I have the following 2-part question as a homework assignment...
Let $F$ be a field and $nin mathbb{Z}_{geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces
begin{align} {0}subset V_1subset cdotssubset V_{n-1}subset V_n=F^nend{align}
such that $dim V_i=i$ for all $iin {1,2,...,n}$. 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!
linear-algebra abstract-algebra schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
$endgroup$
|
show 1 more comment
$begingroup$
I have the following 2-part question as a homework assignment...
Let $F$ be a field and $nin mathbb{Z}_{geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces
begin{align} {0}subset V_1subset cdotssubset V_{n-1}subset V_n=F^nend{align}
such that $dim V_i=i$ for all $iin {1,2,...,n}$. 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!
linear-algebra abstract-algebra schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
$endgroup$
$begingroup$
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.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:49
$begingroup$
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 $ntimes n$ 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.)
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
$begingroup$
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 $ntimes n$ matrices? (5) Is the action transitive? What about free? etc.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
$begingroup$
@ElizabethS.Q.Goodman you're right, I fixed it.
$endgroup$
– jgcello
Apr 7 '17 at 8:11
$begingroup$
Does that help though or are you still stuck?
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
|
show 1 more comment
$begingroup$
I have the following 2-part question as a homework assignment...
Let $F$ be a field and $nin mathbb{Z}_{geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces
begin{align} {0}subset V_1subset cdotssubset V_{n-1}subset V_n=F^nend{align}
such that $dim V_i=i$ for all $iin {1,2,...,n}$. 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!
linear-algebra abstract-algebra schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
$endgroup$
I have the following 2-part question as a homework assignment...
Let $F$ be a field and $nin mathbb{Z}_{geq 1}$. A full flag in the vector space $F^n$ is a chain of subspaces
begin{align} {0}subset V_1subset cdotssubset V_{n-1}subset V_n=F^nend{align}
such that $dim V_i=i$ for all $iin {1,2,...,n}$. 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!
linear-algebra abstract-algebra schubert-calculus
linear-algebra abstract-algebra schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
39.2k63770
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
asked Apr 7 '17 at 7:37
jgcellojgcello
443310
443310
$begingroup$
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.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:49
$begingroup$
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 $ntimes n$ 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.)
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
$begingroup$
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 $ntimes n$ matrices? (5) Is the action transitive? What about free? etc.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
$begingroup$
@ElizabethS.Q.Goodman you're right, I fixed it.
$endgroup$
– jgcello
Apr 7 '17 at 8:11
$begingroup$
Does that help though or are you still stuck?
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
|
show 1 more comment
$begingroup$
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.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:49
$begingroup$
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 $ntimes n$ 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.)
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
$begingroup$
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 $ntimes n$ matrices? (5) Is the action transitive? What about free? etc.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
$begingroup$
@ElizabethS.Q.Goodman you're right, I fixed it.
$endgroup$
– jgcello
Apr 7 '17 at 8:11
$begingroup$
Does that help though or are you still stuck?
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
$begingroup$
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.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:49
$begingroup$
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.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:49
$begingroup$
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 $ntimes n$ 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.)
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
$begingroup$
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 $ntimes n$ 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.)
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
2
$begingroup$
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 $ntimes n$ matrices? (5) Is the action transitive? What about free? etc.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
$begingroup$
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 $ntimes n$ matrices? (5) Is the action transitive? What about free? etc.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
$begingroup$
@ElizabethS.Q.Goodman you're right, I fixed it.
$endgroup$
– jgcello
Apr 7 '17 at 8:11
$begingroup$
@ElizabethS.Q.Goodman you're right, I fixed it.
$endgroup$
– jgcello
Apr 7 '17 at 8:11
$begingroup$
Does that help though or are you still stuck?
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
$begingroup$
Does that help though or are you still stuck?
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
|
show 1 more comment
1 Answer
1
active
oldest
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$begingroup$
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 $begin{pmatrix} lambda \ 0 \ dots \ 0end{pmatrix}$
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 $begin{pmatrix} mu \ gamma \ 0 \ dots \ 0end{pmatrix}$. 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 $F^4$.
$endgroup$
add a comment |
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$begingroup$
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 $begin{pmatrix} lambda \ 0 \ dots \ 0end{pmatrix}$
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 $begin{pmatrix} mu \ gamma \ 0 \ dots \ 0end{pmatrix}$. 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 $F^4$.
$endgroup$
add a comment |
$begingroup$
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 $begin{pmatrix} lambda \ 0 \ dots \ 0end{pmatrix}$
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 $begin{pmatrix} mu \ gamma \ 0 \ dots \ 0end{pmatrix}$. 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 $F^4$.
$endgroup$
add a comment |
$begingroup$
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 $begin{pmatrix} lambda \ 0 \ dots \ 0end{pmatrix}$
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 $begin{pmatrix} mu \ gamma \ 0 \ dots \ 0end{pmatrix}$. 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 $F^4$.
$endgroup$
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 $begin{pmatrix} lambda \ 0 \ dots \ 0end{pmatrix}$
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 $begin{pmatrix} mu \ gamma \ 0 \ dots \ 0end{pmatrix}$. 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 $F^4$.
answered Apr 7 '17 at 10:23
user171326
add a comment |
add a comment |
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$begingroup$
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.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:49
$begingroup$
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 $ntimes n$ 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.)
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
$begingroup$
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 $ntimes n$ matrices? (5) Is the action transitive? What about free? etc.
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
$begingroup$
@ElizabethS.Q.Goodman you're right, I fixed it.
$endgroup$
– jgcello
Apr 7 '17 at 8:11
$begingroup$
Does that help though or are you still stuck?
$endgroup$
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20