Calculating the Formal character on the irreducible $(n+1)$ dimensional representation of $mathfrak{sl}_2$
Let $V(n)$ be the unique, irreducible representation of $mathfrak{sl}_2$ of $(n+1)$-dimensions.
Let $rho$ be the sum of all fundamental weights.
I want to calculate the formal character $ch(V(n)) = sum_{mu in X}(dim(V(n)_mu))e^mu$
Now, I don't think I can apply the Weyl Character Formula because I don't know anything about there being a dominant weight, which makes me think I'm meant to use this formula.
I am asked to give the formal character in terms of $rho$.
My first thought is that $mathfrak{sl}_2$ has a type $A_1$ root system, and so there is one simple root, and one fundamental weight.
I know that both the roots in the $A_1$ root system are in the weight lattice, and including the fundamental weight this gives three weights to consider for the sum.
Additionally, the fact that there is only one fundamental weight, $omega$ means that $rho = omega$ and we also know that $omega = frac{1}{2}alpha$, where $alpha$ is the simple root.
I am a bit stuck now and I'm not really sure how to proceed in calculating the formal character. How do I know which weights matter?
lie-algebras root-systems
asked 2 days ago
user366818user366818
902410
add a comment |
Let $V(n)$ be the unique, irreducible representation of $mathfrak{sl}_2$ of $(n+1)$-dimensions.
Let $rho$ be the sum of all fundamental weights.
I want to calculate the formal character $ch(V(n)) = sum_{mu in X}(dim(V(n)_mu))e^mu$
Now, I don't think I can apply the Weyl Character Formula because I don't know anything about there being a dominant weight, which makes me think I'm meant to use this formula.
I am asked to give the formal character in terms of $rho$.
My first thought is that $mathfrak{sl}_2$ has a type $A_1$ root system, and so there is one simple root, and one fundamental weight.
I know that both the roots in the $A_1$ root system are in the weight lattice, and including the fundamental weight this gives three weights to consider for the sum.
Additionally, the fact that there is only one fundamental weight, $omega$ means that $rho = omega$ and we also know that $omega = frac{1}{2}alpha$, where $alpha$ is the simple root.
I am a bit stuck now and I'm not really sure how to proceed in calculating the formal character. How do I know which weights matter?
lie-algebras root-systems
asked 2 days ago
user366818user366818
902410
add a comment |
Let $V(n)$ be the unique, irreducible representation of $mathfrak{sl}_2$ of $(n+1)$-dimensions.
Let $rho$ be the sum of all fundamental weights.
I want to calculate the formal character $ch(V(n)) = sum_{mu in X}(dim(V(n)_mu))e^mu$
Now, I don't think I can apply the Weyl Character Formula because I don't know anything about there being a dominant weight, which makes me think I'm meant to use this formula.
I am asked to give the formal character in terms of $rho$.
My first thought is that $mathfrak{sl}_2$ has a type $A_1$ root system, and so there is one simple root, and one fundamental weight.
I know that both the roots in the $A_1$ root system are in the weight lattice, and including the fundamental weight this gives three weights to consider for the sum.
Additionally, the fact that there is only one fundamental weight, $omega$ means that $rho = omega$ and we also know that $omega = frac{1}{2}alpha$, where $alpha$ is the simple root.
I am a bit stuck now and I'm not really sure how to proceed in calculating the formal character. How do I know which weights matter?
lie-algebras root-systems
asked 2 days ago
user366818user366818
902410
Let $V(n)$ be the unique, irreducible representation of $mathfrak{sl}_2$ of $(n+1)$-dimensions.
Let $rho$ be the sum of all fundamental weights.
I want to calculate the formal character $ch(V(n)) = sum_{mu in X}(dim(V(n)_mu))e^mu$
Now, I don't think I can apply the Weyl Character Formula because I don't know anything about there being a dominant weight, which makes me think I'm meant to use this formula.
I am asked to give the formal character in terms of $rho$.
My first thought is that $mathfrak{sl}_2$ has a type $A_1$ root system, and so there is one simple root, and one fundamental weight.
I know that both the roots in the $A_1$ root system are in the weight lattice, and including the fundamental weight this gives three weights to consider for the sum.
Additionally, the fact that there is only one fundamental weight, $omega$ means that $rho = omega$ and we also know that $omega = frac{1}{2}alpha$, where $alpha$ is the simple root.
I am a bit stuck now and I'm not really sure how to proceed in calculating the formal character. How do I know which weights matter?
lie-algebras root-systems
lie-algebras root-systems
asked 2 days ago
user366818user366818
902410
asked 2 days ago
user366818user366818
902410
asked 2 days ago
user366818user366818
902410
asked 2 days ago
user366818user366818
902410
asked 2 days ago
user366818user366818
902410
902410
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%2f3063050%2fcalculating-the-formal-character-on-the-irreducible-n1-dimensional-represen%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.
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%2f3063050%2fcalculating-the-formal-character-on-the-irreducible-n1-dimensional-represen%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
