$GL_n(F)$ acts on the flag variety
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 yesterday
Matt Samuel
37k63465
asked Apr 7 '17 at 7:37
jgcello
433210
|
show 1 more comment
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 yesterday
Matt Samuel
37k63465
asked Apr 7 '17 at 7:37
jgcello
433210
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 7 '17 at 7:49
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 $ntimes n$ 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.)
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
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.
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
@ElizabethS.Q.Goodman you're right, I fixed it.
– jgcello
Apr 7 '17 at 8:11
Does that help though or are you still stuck?
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
|
show 1 more comment
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 yesterday
Matt Samuel
37k63465
asked Apr 7 '17 at 7:37
jgcello
433210
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 yesterday
Matt Samuel
37k63465
asked Apr 7 '17 at 7:37
jgcello
433210
edited yesterday
Matt Samuel
37k63465
asked Apr 7 '17 at 7:37
jgcello
433210
edited yesterday
Matt Samuel
37k63465
edited yesterday
Matt Samuel
37k63465
edited yesterday
Matt Samuel
37k63465
37k63465
asked Apr 7 '17 at 7:37
jgcello
433210
asked Apr 7 '17 at 7:37
jgcello
433210
asked Apr 7 '17 at 7:37
jgcello
433210
433210
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 7 '17 at 7:49
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 $ntimes n$ 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.)
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
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.
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
@ElizabethS.Q.Goodman you're right, I fixed it.
– jgcello
Apr 7 '17 at 8:11
Does that help though or are you still stuck?
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
|
show 1 more comment
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 7 '17 at 7:49
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 $ntimes n$ 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.)
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
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.
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
@ElizabethS.Q.Goodman you're right, I fixed it.
– jgcello
Apr 7 '17 at 8:11
Does that help though or are you still stuck?
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
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 7 '17 at 7:49
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 7 '17 at 7:49
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 $ntimes n$ 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.)
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
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.)
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
2
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.
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
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.
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
@ElizabethS.Q.Goodman you're right, I fixed it.
– jgcello
Apr 7 '17 at 8:11
@ElizabethS.Q.Goodman you're right, I fixed it.
– jgcello
Apr 7 '17 at 8:11
Does that help though or are you still stuck?
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
Does that help though or are you still stuck?
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20
|
show 1 more comment
1 Answer
1
active
oldest
votes
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 $F^4$.
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2222023%2fgl-nf-acts-on-the-flag-variety%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 $F^4$.
add a comment |
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 $F^4$.
add a comment |
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 $F^4$.
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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2222023%2fgl-nf-acts-on-the-flag-variety%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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 7 '17 at 7:49
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 $ntimes n$ 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.)
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:53
2
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.
– Elizabeth S. Q. Goodman
Apr 7 '17 at 7:55
@ElizabethS.Q.Goodman you're right, I fixed it.
– jgcello
Apr 7 '17 at 8:11
Does that help though or are you still stuck?
– Elizabeth S. Q. Goodman
Apr 7 '17 at 8:20