Mixing
Problems for a concrete example in representation theory
$begingroup$
I am currently studying representation theory. In my courses notes, I face trouble understanding one step of a concrete example: three masses connected by springs. This is the same approach as in Georgi p27 http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf
My problem happen when we have to find the normal modes associated to $D_2$. This point is not fully developped in Georgi. However, in my courses notes, I find this explanation:
"Since $D_6 = D^{def} otimes D_2$ and that $D^{def} = D_1 oplus D_2$, we have
$D_6 = [D_1 oplus D_2] otimes D_2 = (D_2 otimes D_2) oplus (D_2 otimes D_2) = D_2 oplus ( D_2 otimes D_2)$.
We can then see that the direction in $mathbb{R}^3$ invariant under $D^{def}$ tensorized with $mathbb{R}^2$ transforms under $D_2$."
From this fact, he deduce that $(1,0,1,0,1,0)$ for instance must be a normal mode. I understand all of that except the last sentence: Why is this direction transforming like $D_2$ ? ( I guess transforms under ... mean is an invariant subspace under ... ).
Thanks.
representation-theory physics
asked Jan 25 at 21:30
thephysics17thephysics17
162
$endgroup$
add a comment |
$begingroup$
I am currently studying representation theory. In my courses notes, I face trouble understanding one step of a concrete example: three masses connected by springs. This is the same approach as in Georgi p27 http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf
My problem happen when we have to find the normal modes associated to $D_2$. This point is not fully developped in Georgi. However, in my courses notes, I find this explanation:
"Since $D_6 = D^{def} otimes D_2$ and that $D^{def} = D_1 oplus D_2$, we have
$D_6 = [D_1 oplus D_2] otimes D_2 = (D_2 otimes D_2) oplus (D_2 otimes D_2) = D_2 oplus ( D_2 otimes D_2)$.
We can then see that the direction in $mathbb{R}^3$ invariant under $D^{def}$ tensorized with $mathbb{R}^2$ transforms under $D_2$."
From this fact, he deduce that $(1,0,1,0,1,0)$ for instance must be a normal mode. I understand all of that except the last sentence: Why is this direction transforming like $D_2$ ? ( I guess transforms under ... mean is an invariant subspace under ... ).
Thanks.
representation-theory physics
asked Jan 25 at 21:30
thephysics17thephysics17
162
$endgroup$
add a comment |
$begingroup$
I am currently studying representation theory. In my courses notes, I face trouble understanding one step of a concrete example: three masses connected by springs. This is the same approach as in Georgi p27 http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf
My problem happen when we have to find the normal modes associated to $D_2$. This point is not fully developped in Georgi. However, in my courses notes, I find this explanation:
"Since $D_6 = D^{def} otimes D_2$ and that $D^{def} = D_1 oplus D_2$, we have
$D_6 = [D_1 oplus D_2] otimes D_2 = (D_2 otimes D_2) oplus (D_2 otimes D_2) = D_2 oplus ( D_2 otimes D_2)$.
We can then see that the direction in $mathbb{R}^3$ invariant under $D^{def}$ tensorized with $mathbb{R}^2$ transforms under $D_2$."
From this fact, he deduce that $(1,0,1,0,1,0)$ for instance must be a normal mode. I understand all of that except the last sentence: Why is this direction transforming like $D_2$ ? ( I guess transforms under ... mean is an invariant subspace under ... ).
Thanks.
representation-theory physics
asked Jan 25 at 21:30
thephysics17thephysics17
162
$endgroup$
I am currently studying representation theory. In my courses notes, I face trouble understanding one step of a concrete example: three masses connected by springs. This is the same approach as in Georgi p27 http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf
My problem happen when we have to find the normal modes associated to $D_2$. This point is not fully developped in Georgi. However, in my courses notes, I find this explanation:
"Since $D_6 = D^{def} otimes D_2$ and that $D^{def} = D_1 oplus D_2$, we have
$D_6 = [D_1 oplus D_2] otimes D_2 = (D_2 otimes D_2) oplus (D_2 otimes D_2) = D_2 oplus ( D_2 otimes D_2)$.
We can then see that the direction in $mathbb{R}^3$ invariant under $D^{def}$ tensorized with $mathbb{R}^2$ transforms under $D_2$."
From this fact, he deduce that $(1,0,1,0,1,0)$ for instance must be a normal mode. I understand all of that except the last sentence: Why is this direction transforming like $D_2$ ? ( I guess transforms under ... mean is an invariant subspace under ... ).
Thanks.
representation-theory physics
representation-theory physics
asked Jan 25 at 21:30
thephysics17thephysics17
162
asked Jan 25 at 21:30
thephysics17thephysics17
162
asked Jan 25 at 21:30
thephysics17thephysics17
162
asked Jan 25 at 21:30
thephysics17thephysics17
162
asked Jan 25 at 21:30
thephysics17thephysics17
162
162
add a comment |
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%2f3087642%2fproblems-for-a-concrete-example-in-representation-theory%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%2f3087642%2fproblems-for-a-concrete-example-in-representation-theory%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
