How exactly does character inflation work?
$begingroup$
Let $A_4$ be the alternating group and let $V = {{(1), (12)(34), (13)(24), (14)(23)}}$ be a normal subgroup of $A_4$. Then $A_4/V simeq C_3$, so $A_4$ has $3$ one dimensional representations, which apparently, we can inflate/lift from the character table for $C_3$.
I don't really understand how this works? How do I relate the character values of $C_3$ to $A_4$? Do I need to define a homomorphism from $A_4$ to $C_3$?
abstract-algebra representation-theory characters
asked Jan 21 at 21:37
the manthe man
772715
$endgroup$
add a comment |
$begingroup$
Let $A_4$ be the alternating group and let $V = {{(1), (12)(34), (13)(24), (14)(23)}}$ be a normal subgroup of $A_4$. Then $A_4/V simeq C_3$, so $A_4$ has $3$ one dimensional representations, which apparently, we can inflate/lift from the character table for $C_3$.
I don't really understand how this works? How do I relate the character values of $C_3$ to $A_4$? Do I need to define a homomorphism from $A_4$ to $C_3$?
abstract-algebra representation-theory characters
asked Jan 21 at 21:37
the manthe man
772715
$endgroup$
$begingroup$
Just look at the representation $A_4 to A_4 /V to GL(X)$ induced from a representation $A_4/V to GL(X)$.
$endgroup$
– anomaly
Jan 21 at 22:00
add a comment |
$begingroup$
Let $A_4$ be the alternating group and let $V = {{(1), (12)(34), (13)(24), (14)(23)}}$ be a normal subgroup of $A_4$. Then $A_4/V simeq C_3$, so $A_4$ has $3$ one dimensional representations, which apparently, we can inflate/lift from the character table for $C_3$.
I don't really understand how this works? How do I relate the character values of $C_3$ to $A_4$? Do I need to define a homomorphism from $A_4$ to $C_3$?
abstract-algebra representation-theory characters
asked Jan 21 at 21:37
the manthe man
772715
$endgroup$
Let $A_4$ be the alternating group and let $V = {{(1), (12)(34), (13)(24), (14)(23)}}$ be a normal subgroup of $A_4$. Then $A_4/V simeq C_3$, so $A_4$ has $3$ one dimensional representations, which apparently, we can inflate/lift from the character table for $C_3$.
I don't really understand how this works? How do I relate the character values of $C_3$ to $A_4$? Do I need to define a homomorphism from $A_4$ to $C_3$?
abstract-algebra representation-theory characters
abstract-algebra representation-theory characters
asked Jan 21 at 21:37
the manthe man
772715
asked Jan 21 at 21:37
the manthe man
772715
asked Jan 21 at 21:37
the manthe man
772715
asked Jan 21 at 21:37
the manthe man
772715
asked Jan 21 at 21:37
the manthe man
772715
772715
$begingroup$
Just look at the representation $A_4 to A_4 /V to GL(X)$ induced from a representation $A_4/V to GL(X)$.
$endgroup$
– anomaly
Jan 21 at 22:00
add a comment |
$begingroup$
Just look at the representation $A_4 to A_4 /V to GL(X)$ induced from a representation $A_4/V to GL(X)$.
$endgroup$
– anomaly
Jan 21 at 22:00
$begingroup$
Just look at the representation $A_4 to A_4 /V to GL(X)$ induced from a representation $A_4/V to GL(X)$.
$endgroup$
– anomaly
Jan 21 at 22:00
$begingroup$
Just look at the representation $A_4 to A_4 /V to GL(X)$ induced from a representation $A_4/V to GL(X)$.
$endgroup$
– anomaly
Jan 21 at 22:00
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%2f3082451%2fhow-exactly-does-character-inflation-work%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%2f3082451%2fhow-exactly-does-character-inflation-work%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
$begingroup$
Just look at the representation $A_4 to A_4 /V to GL(X)$ induced from a representation $A_4/V to GL(X)$.
$endgroup$
– anomaly
Jan 21 at 22:00