Is there any relation between real and complex character functions of irreducible representations of compact...
$begingroup$
Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $chi_U^mathbb{R}:Gtomathbb{R}$ as $chi_U^mathbb{R}(g)=operatorname{Tr}(l_g)$. If $V$ is a complex $G$-module one can define its character as $chi_V:Gtomathbb{C}$ as $chi_V(g)=operatorname{Tr}(l_g)$. Also, one can also define a extension map $e_+(U)=mathbb{C}otimes_mathbb{R} U$ that maps a real $G$-module into a complex $G$-module. Is there any relation between $chi_U^mathbb{R}$ and $chi_{e_+(U)}$ for irreducible $U$?
representation-theory characters
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
$endgroup$
add a comment |
$begingroup$
Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $chi_U^mathbb{R}:Gtomathbb{R}$ as $chi_U^mathbb{R}(g)=operatorname{Tr}(l_g)$. If $V$ is a complex $G$-module one can define its character as $chi_V:Gtomathbb{C}$ as $chi_V(g)=operatorname{Tr}(l_g)$. Also, one can also define a extension map $e_+(U)=mathbb{C}otimes_mathbb{R} U$ that maps a real $G$-module into a complex $G$-module. Is there any relation between $chi_U^mathbb{R}$ and $chi_{e_+(U)}$ for irreducible $U$?
representation-theory characters
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
$endgroup$
1
$begingroup$
If $E$ is a real vector space then $ mathbb{C} otimes_{mathbb{R}}E =1 otimes_{mathbb{R}}E+iotimes_{mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $mathbb{R}$-linear maps $E to mathbb{R}$ have a natural $mathbb{C}$-linear extension $mathbb{C} otimes_{mathbb{R}}E to mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) to mathbb{R}$.
$endgroup$
– reuns
Jan 15 at 0:31
$begingroup$
But $End(Cotimes_mathbb{R} U)=Cotimes_mathbb{R} End(U)$?
$endgroup$
– Andre Gomes
Jan 15 at 3:35
$begingroup$
$End_k(V)$ : the $k$-linear maps $V to V$
$endgroup$
– reuns
Jan 15 at 4:07
add a comment |
$begingroup$
Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $chi_U^mathbb{R}:Gtomathbb{R}$ as $chi_U^mathbb{R}(g)=operatorname{Tr}(l_g)$. If $V$ is a complex $G$-module one can define its character as $chi_V:Gtomathbb{C}$ as $chi_V(g)=operatorname{Tr}(l_g)$. Also, one can also define a extension map $e_+(U)=mathbb{C}otimes_mathbb{R} U$ that maps a real $G$-module into a complex $G$-module. Is there any relation between $chi_U^mathbb{R}$ and $chi_{e_+(U)}$ for irreducible $U$?
representation-theory characters
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
$endgroup$
Let $G$ be a compact lie group and $U$ a real $G$-module. One can define the real character as $chi_U^mathbb{R}:Gtomathbb{R}$ as $chi_U^mathbb{R}(g)=operatorname{Tr}(l_g)$. If $V$ is a complex $G$-module one can define its character as $chi_V:Gtomathbb{C}$ as $chi_V(g)=operatorname{Tr}(l_g)$. Also, one can also define a extension map $e_+(U)=mathbb{C}otimes_mathbb{R} U$ that maps a real $G$-module into a complex $G$-module. Is there any relation between $chi_U^mathbb{R}$ and $chi_{e_+(U)}$ for irreducible $U$?
representation-theory characters
representation-theory characters
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
asked Jan 14 at 22:47
Andre GomesAndre Gomes
930516
930516
1
$begingroup$
If $E$ is a real vector space then $ mathbb{C} otimes_{mathbb{R}}E =1 otimes_{mathbb{R}}E+iotimes_{mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $mathbb{R}$-linear maps $E to mathbb{R}$ have a natural $mathbb{C}$-linear extension $mathbb{C} otimes_{mathbb{R}}E to mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) to mathbb{R}$.
$endgroup$
– reuns
Jan 15 at 0:31
$begingroup$
But $End(Cotimes_mathbb{R} U)=Cotimes_mathbb{R} End(U)$?
$endgroup$
– Andre Gomes
Jan 15 at 3:35
$begingroup$
$End_k(V)$ : the $k$-linear maps $V to V$
$endgroup$
– reuns
Jan 15 at 4:07
add a comment |
1
$begingroup$
If $E$ is a real vector space then $ mathbb{C} otimes_{mathbb{R}}E =1 otimes_{mathbb{R}}E+iotimes_{mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $mathbb{R}$-linear maps $E to mathbb{R}$ have a natural $mathbb{C}$-linear extension $mathbb{C} otimes_{mathbb{R}}E to mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) to mathbb{R}$.
$endgroup$
– reuns
Jan 15 at 0:31
$begingroup$
But $End(Cotimes_mathbb{R} U)=Cotimes_mathbb{R} End(U)$?
$endgroup$
– Andre Gomes
Jan 15 at 3:35
$begingroup$
$End_k(V)$ : the $k$-linear maps $V to V$
$endgroup$
– reuns
Jan 15 at 4:07
1
1
$begingroup$
If $E$ is a real vector space then $ mathbb{C} otimes_{mathbb{R}}E =1 otimes_{mathbb{R}}E+iotimes_{mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $mathbb{R}$-linear maps $E to mathbb{R}$ have a natural $mathbb{C}$-linear extension $mathbb{C} otimes_{mathbb{R}}E to mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) to mathbb{R}$.
$endgroup$
– reuns
Jan 15 at 0:31
$begingroup$
If $E$ is a real vector space then $ mathbb{C} otimes_{mathbb{R}}E =1 otimes_{mathbb{R}}E+iotimes_{mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $mathbb{R}$-linear maps $E to mathbb{R}$ have a natural $mathbb{C}$-linear extension $mathbb{C} otimes_{mathbb{R}}E to mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) to mathbb{R}$.
$endgroup$
– reuns
Jan 15 at 0:31
$begingroup$
But $End(Cotimes_mathbb{R} U)=Cotimes_mathbb{R} End(U)$?
$endgroup$
– Andre Gomes
Jan 15 at 3:35
$begingroup$
But $End(Cotimes_mathbb{R} U)=Cotimes_mathbb{R} End(U)$?
$endgroup$
– Andre Gomes
Jan 15 at 3:35
$begingroup$
$End_k(V)$ : the $k$-linear maps $V to V$
$endgroup$
– reuns
Jan 15 at 4:07
$begingroup$
$End_k(V)$ : the $k$-linear maps $V to V$
$endgroup$
– reuns
Jan 15 at 4:07
add a comment |
0
active
oldest
votes
Your Answer
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%2f3073857%2fis-there-any-relation-between-real-and-complex-character-functions-of-irreducibl%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%2f3073857%2fis-there-any-relation-between-real-and-complex-character-functions-of-irreducibl%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
1
$begingroup$
If $E$ is a real vector space then $ mathbb{C} otimes_{mathbb{R}}E =1 otimes_{mathbb{R}}E+iotimes_{mathbb{R}} E =E+iE$ where $(a+ib)(u+iv) = au-bv+i(bu+av)$ and the $mathbb{R}$-linear maps $E to mathbb{R}$ have a natural $mathbb{C}$-linear extension $mathbb{C} otimes_{mathbb{R}}E to mathbb{C}$ (extension means they stay the same on $E$). Here $E = End(U)$ and $Tr : End(U) to mathbb{R}$.
$endgroup$
– reuns
Jan 15 at 0:31
$begingroup$
But $End(Cotimes_mathbb{R} U)=Cotimes_mathbb{R} End(U)$?
$endgroup$
– Andre Gomes
Jan 15 at 3:35
$begingroup$
$End_k(V)$ : the $k$-linear maps $V to V$
$endgroup$
– reuns
Jan 15 at 4:07