Proof of Schur polynomials relation equality
$begingroup$
Let $ lambda $ be a partition with $l(lambda) leq m$ and $lambda_1 leq n$. We define
$$ hatlambda = (n - lambda_{m}, n - lambda_{m-1}, dots, n - lambda_{1}).$$
What is the prove for the following two equalities?
$a)$
$$(x_1x_2 cdots x_m)^ns_lambda(x^{-1}_1, dots,x^{-1}_m ) = s_hatlambda(x_1, dots,x_m )$$
$b)$
$$prod_{i=1}^m prod_{j=1}^n(x_i + y_j) = sum_lambda s_lambda (x_1, dots,x_m )s_hatlambda(y_1, dots,y_n )$$
The last sum is over all partitions $lambda$ with $l(lambda) leq m$ and $lambda_1 leq n$ and
$s_lambda$ denotes Schur polinomial.
proof-verification polynomials schur-complement
asked Dec 20 '18 at 18:53
thinker123thinker123
12
$endgroup$
add a comment |
$begingroup$
Let $ lambda $ be a partition with $l(lambda) leq m$ and $lambda_1 leq n$. We define
$$ hatlambda = (n - lambda_{m}, n - lambda_{m-1}, dots, n - lambda_{1}).$$
What is the prove for the following two equalities?
$a)$
$$(x_1x_2 cdots x_m)^ns_lambda(x^{-1}_1, dots,x^{-1}_m ) = s_hatlambda(x_1, dots,x_m )$$
$b)$
$$prod_{i=1}^m prod_{j=1}^n(x_i + y_j) = sum_lambda s_lambda (x_1, dots,x_m )s_hatlambda(y_1, dots,y_n )$$
The last sum is over all partitions $lambda$ with $l(lambda) leq m$ and $lambda_1 leq n$ and
$s_lambda$ denotes Schur polinomial.
proof-verification polynomials schur-complement
asked Dec 20 '18 at 18:53
thinker123thinker123
12
$endgroup$
add a comment |
$begingroup$
Let $ lambda $ be a partition with $l(lambda) leq m$ and $lambda_1 leq n$. We define
$$ hatlambda = (n - lambda_{m}, n - lambda_{m-1}, dots, n - lambda_{1}).$$
What is the prove for the following two equalities?
$a)$
$$(x_1x_2 cdots x_m)^ns_lambda(x^{-1}_1, dots,x^{-1}_m ) = s_hatlambda(x_1, dots,x_m )$$
$b)$
$$prod_{i=1}^m prod_{j=1}^n(x_i + y_j) = sum_lambda s_lambda (x_1, dots,x_m )s_hatlambda(y_1, dots,y_n )$$
The last sum is over all partitions $lambda$ with $l(lambda) leq m$ and $lambda_1 leq n$ and
$s_lambda$ denotes Schur polinomial.
proof-verification polynomials schur-complement
asked Dec 20 '18 at 18:53
thinker123thinker123
12
$endgroup$
Let $ lambda $ be a partition with $l(lambda) leq m$ and $lambda_1 leq n$. We define
$$ hatlambda = (n - lambda_{m}, n - lambda_{m-1}, dots, n - lambda_{1}).$$
What is the prove for the following two equalities?
$a)$
$$(x_1x_2 cdots x_m)^ns_lambda(x^{-1}_1, dots,x^{-1}_m ) = s_hatlambda(x_1, dots,x_m )$$
$b)$
$$prod_{i=1}^m prod_{j=1}^n(x_i + y_j) = sum_lambda s_lambda (x_1, dots,x_m )s_hatlambda(y_1, dots,y_n )$$
The last sum is over all partitions $lambda$ with $l(lambda) leq m$ and $lambda_1 leq n$ and
$s_lambda$ denotes Schur polinomial.
proof-verification polynomials schur-complement
proof-verification polynomials schur-complement
asked Dec 20 '18 at 18:53
thinker123thinker123
12
asked Dec 20 '18 at 18:53
thinker123thinker123
12
asked Dec 20 '18 at 18:53
thinker123thinker123
12
asked Dec 20 '18 at 18:53
thinker123thinker123
12
asked Dec 20 '18 at 18:53
thinker123thinker123
12
12
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%2f3047859%2fproof-of-schur-polynomials-relation-equality%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%2f3047859%2fproof-of-schur-polynomials-relation-equality%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
