Bruhat decomposition of flag variety.
$begingroup$
Let $GL_{n}/B_{n}$ be the complete flag variety (where $B_{n}$ is the group of upper-triangular nonsingular matrices). Let $W$ be the Weyl group of $U(n)$ (that in this contest is a permutation group). We have $GL_{n}/B_{n} simeq U(n)/T_{n}$ (where $U(n)$ is the unitary group and $T_{n}$ its maximal torus). I have to build an explicit Bruhat decomposition of $U(n)/T_{n}$. Could you help me to find the cells end prove that there are $|W|$ cells of even dimension? (Also in a particular situation, for example $U(3)/(S^{1})^{3}$.)
abstract-algebra algebraic-topology schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Jan 16 '13 at 11:54
user58359user58359
311
$endgroup$
add a comment |
$begingroup$
Let $GL_{n}/B_{n}$ be the complete flag variety (where $B_{n}$ is the group of upper-triangular nonsingular matrices). Let $W$ be the Weyl group of $U(n)$ (that in this contest is a permutation group). We have $GL_{n}/B_{n} simeq U(n)/T_{n}$ (where $U(n)$ is the unitary group and $T_{n}$ its maximal torus). I have to build an explicit Bruhat decomposition of $U(n)/T_{n}$. Could you help me to find the cells end prove that there are $|W|$ cells of even dimension? (Also in a particular situation, for example $U(3)/(S^{1})^{3}$.)
abstract-algebra algebraic-topology schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Jan 16 '13 at 11:54
user58359user58359
311
$endgroup$
2
$begingroup$
How is this different from your previous question: math.stackexchange.com/questions/279270/…? If not substantially different, you may consider deleting this one and editing the previous one.
$endgroup$
– Jason DeVito
Jan 16 '13 at 12:58
$begingroup$
Aren't these cells essentially subsets of the ${n-1choose 2}$ matrix entries below the diagonal? We do require that the lower triangular matrices with non-zero entries limited to the subset form a subgroup, or equivalently, the subset must meet the condition that position $(i,j), i>j,$ is included if any position on row $i$ AND any position on column $j$ is included. When $n=3$ the "legal" sets are: the empty set, ${(2,1)}$, ${3,2}$, ${(3,1),(3,2)}$, ${(2,1),(3,1)}$ and ${(2,1),(3,1),(3,2)}$. Each position can house a complex number so a cell with $j$ positions has dimension $2j$.
$endgroup$
– Jyrki Lahtonen
Jan 16 '13 at 13:19
add a comment |
$begingroup$
Let $GL_{n}/B_{n}$ be the complete flag variety (where $B_{n}$ is the group of upper-triangular nonsingular matrices). Let $W$ be the Weyl group of $U(n)$ (that in this contest is a permutation group). We have $GL_{n}/B_{n} simeq U(n)/T_{n}$ (where $U(n)$ is the unitary group and $T_{n}$ its maximal torus). I have to build an explicit Bruhat decomposition of $U(n)/T_{n}$. Could you help me to find the cells end prove that there are $|W|$ cells of even dimension? (Also in a particular situation, for example $U(3)/(S^{1})^{3}$.)
abstract-algebra algebraic-topology schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Jan 16 '13 at 11:54
user58359user58359
311
$endgroup$
Let $GL_{n}/B_{n}$ be the complete flag variety (where $B_{n}$ is the group of upper-triangular nonsingular matrices). Let $W$ be the Weyl group of $U(n)$ (that in this contest is a permutation group). We have $GL_{n}/B_{n} simeq U(n)/T_{n}$ (where $U(n)$ is the unitary group and $T_{n}$ its maximal torus). I have to build an explicit Bruhat decomposition of $U(n)/T_{n}$. Could you help me to find the cells end prove that there are $|W|$ cells of even dimension? (Also in a particular situation, for example $U(3)/(S^{1})^{3}$.)
abstract-algebra algebraic-topology schubert-calculus
abstract-algebra algebraic-topology schubert-calculus
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Jan 16 '13 at 11:54
user58359user58359
311
edited Dec 25 '18 at 11:45
Matt Samuel
39.2k63770
asked Jan 16 '13 at 11:54
user58359user58359
311
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 Jan 16 '13 at 11:54
user58359user58359
311
asked Jan 16 '13 at 11:54
user58359user58359
311
asked Jan 16 '13 at 11:54
user58359user58359
311
311
2
$begingroup$
How is this different from your previous question: math.stackexchange.com/questions/279270/…? If not substantially different, you may consider deleting this one and editing the previous one.
$endgroup$
– Jason DeVito
Jan 16 '13 at 12:58
$begingroup$
Aren't these cells essentially subsets of the ${n-1choose 2}$ matrix entries below the diagonal? We do require that the lower triangular matrices with non-zero entries limited to the subset form a subgroup, or equivalently, the subset must meet the condition that position $(i,j), i>j,$ is included if any position on row $i$ AND any position on column $j$ is included. When $n=3$ the "legal" sets are: the empty set, ${(2,1)}$, ${3,2}$, ${(3,1),(3,2)}$, ${(2,1),(3,1)}$ and ${(2,1),(3,1),(3,2)}$. Each position can house a complex number so a cell with $j$ positions has dimension $2j$.
$endgroup$
– Jyrki Lahtonen
Jan 16 '13 at 13:19
add a comment |
2
$begingroup$
How is this different from your previous question: math.stackexchange.com/questions/279270/…? If not substantially different, you may consider deleting this one and editing the previous one.
$endgroup$
– Jason DeVito
Jan 16 '13 at 12:58
$begingroup$
Aren't these cells essentially subsets of the ${n-1choose 2}$ matrix entries below the diagonal? We do require that the lower triangular matrices with non-zero entries limited to the subset form a subgroup, or equivalently, the subset must meet the condition that position $(i,j), i>j,$ is included if any position on row $i$ AND any position on column $j$ is included. When $n=3$ the "legal" sets are: the empty set, ${(2,1)}$, ${3,2}$, ${(3,1),(3,2)}$, ${(2,1),(3,1)}$ and ${(2,1),(3,1),(3,2)}$. Each position can house a complex number so a cell with $j$ positions has dimension $2j$.
$endgroup$
– Jyrki Lahtonen
Jan 16 '13 at 13:19
2
2
$begingroup$
How is this different from your previous question: math.stackexchange.com/questions/279270/…? If not substantially different, you may consider deleting this one and editing the previous one.
$endgroup$
– Jason DeVito
Jan 16 '13 at 12:58
$begingroup$
How is this different from your previous question: math.stackexchange.com/questions/279270/…? If not substantially different, you may consider deleting this one and editing the previous one.
$endgroup$
– Jason DeVito
Jan 16 '13 at 12:58
$begingroup$
Aren't these cells essentially subsets of the ${n-1choose 2}$ matrix entries below the diagonal? We do require that the lower triangular matrices with non-zero entries limited to the subset form a subgroup, or equivalently, the subset must meet the condition that position $(i,j), i>j,$ is included if any position on row $i$ AND any position on column $j$ is included. When $n=3$ the "legal" sets are: the empty set, ${(2,1)}$, ${3,2}$, ${(3,1),(3,2)}$, ${(2,1),(3,1)}$ and ${(2,1),(3,1),(3,2)}$. Each position can house a complex number so a cell with $j$ positions has dimension $2j$.
$endgroup$
– Jyrki Lahtonen
Jan 16 '13 at 13:19
$begingroup$
Aren't these cells essentially subsets of the ${n-1choose 2}$ matrix entries below the diagonal? We do require that the lower triangular matrices with non-zero entries limited to the subset form a subgroup, or equivalently, the subset must meet the condition that position $(i,j), i>j,$ is included if any position on row $i$ AND any position on column $j$ is included. When $n=3$ the "legal" sets are: the empty set, ${(2,1)}$, ${3,2}$, ${(3,1),(3,2)}$, ${(2,1),(3,1)}$ and ${(2,1),(3,1),(3,2)}$. Each position can house a complex number so a cell with $j$ positions has dimension $2j$.
$endgroup$
– Jyrki Lahtonen
Jan 16 '13 at 13:19
add a comment |
0
active
oldest
votes
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%2f279997%2fbruhat-decomposition-of-flag-variety%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f279997%2fbruhat-decomposition-of-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
2
$begingroup$
How is this different from your previous question: math.stackexchange.com/questions/279270/…? If not substantially different, you may consider deleting this one and editing the previous one.
$endgroup$
– Jason DeVito
Jan 16 '13 at 12:58
$begingroup$
Aren't these cells essentially subsets of the ${n-1choose 2}$ matrix entries below the diagonal? We do require that the lower triangular matrices with non-zero entries limited to the subset form a subgroup, or equivalently, the subset must meet the condition that position $(i,j), i>j,$ is included if any position on row $i$ AND any position on column $j$ is included. When $n=3$ the "legal" sets are: the empty set, ${(2,1)}$, ${3,2}$, ${(3,1),(3,2)}$, ${(2,1),(3,1)}$ and ${(2,1),(3,1),(3,2)}$. Each position can house a complex number so a cell with $j$ positions has dimension $2j$.
$endgroup$
– Jyrki Lahtonen
Jan 16 '13 at 13:19