Decomposition of product of two Plucker coordinates
$begingroup$
Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. Then $Gr(k,n)$ is a projective variety and it has Plucker coordinates $P_{i_1, ldots, i_k}$ ($i_1<ldots<i_k$) which is the determinant of the matrix $(x_{ij})_{i in [k], j in {i_1, ldots, i_k}}$. Certain Plucker coordinates satisfy the Plucker relation. For example, for $Gr(2,n)$, $P_{12}P_{34} + P_{23}P_{14}-P_{13}P_{24}=0$. Therefore $P_{13}P_{24} = P_{12}P_{34} + P_{23}P_{14}$ can be viewed as a decomposition of the product of $P_{13}$ and $P_{24}$. For $P_{12}$, $P_{34}$, we say that their product is irreducible. That is $P_{12}P_{34}$ cannot be written as a sum (with two or more terms in the summation, each summand has positive coefficient) of products of Plucker coordinates.
Given two Plucker coordinates $P_{i_1, ldots, i_k}$, $P_{j_1, ldots, j_k}$, is there some formula for the decomposition of $P_{i_1, ldots, i_k} P_{j_1, ldots, j_k} = sum_T c_T P_T$ (P_T is a product of certain Plucker coordinates, $c_T>0$) in the literature? Thank you very much.
ag.algebraic-geometry co.combinatorics
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
$endgroup$
add a comment |
$begingroup$
Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. Then $Gr(k,n)$ is a projective variety and it has Plucker coordinates $P_{i_1, ldots, i_k}$ ($i_1<ldots<i_k$) which is the determinant of the matrix $(x_{ij})_{i in [k], j in {i_1, ldots, i_k}}$. Certain Plucker coordinates satisfy the Plucker relation. For example, for $Gr(2,n)$, $P_{12}P_{34} + P_{23}P_{14}-P_{13}P_{24}=0$. Therefore $P_{13}P_{24} = P_{12}P_{34} + P_{23}P_{14}$ can be viewed as a decomposition of the product of $P_{13}$ and $P_{24}$. For $P_{12}$, $P_{34}$, we say that their product is irreducible. That is $P_{12}P_{34}$ cannot be written as a sum (with two or more terms in the summation, each summand has positive coefficient) of products of Plucker coordinates.
Given two Plucker coordinates $P_{i_1, ldots, i_k}$, $P_{j_1, ldots, j_k}$, is there some formula for the decomposition of $P_{i_1, ldots, i_k} P_{j_1, ldots, j_k} = sum_T c_T P_T$ (P_T is a product of certain Plucker coordinates, $c_T>0$) in the literature? Thank you very much.
ag.algebraic-geometry co.combinatorics
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
$endgroup$
add a comment |
$begingroup$
Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. Then $Gr(k,n)$ is a projective variety and it has Plucker coordinates $P_{i_1, ldots, i_k}$ ($i_1<ldots<i_k$) which is the determinant of the matrix $(x_{ij})_{i in [k], j in {i_1, ldots, i_k}}$. Certain Plucker coordinates satisfy the Plucker relation. For example, for $Gr(2,n)$, $P_{12}P_{34} + P_{23}P_{14}-P_{13}P_{24}=0$. Therefore $P_{13}P_{24} = P_{12}P_{34} + P_{23}P_{14}$ can be viewed as a decomposition of the product of $P_{13}$ and $P_{24}$. For $P_{12}$, $P_{34}$, we say that their product is irreducible. That is $P_{12}P_{34}$ cannot be written as a sum (with two or more terms in the summation, each summand has positive coefficient) of products of Plucker coordinates.
Given two Plucker coordinates $P_{i_1, ldots, i_k}$, $P_{j_1, ldots, j_k}$, is there some formula for the decomposition of $P_{i_1, ldots, i_k} P_{j_1, ldots, j_k} = sum_T c_T P_T$ (P_T is a product of certain Plucker coordinates, $c_T>0$) in the literature? Thank you very much.
ag.algebraic-geometry co.combinatorics
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
$endgroup$
Let $Gr(k,n)$ be the set of all $k$-dimensional subspaces of an $n$-dimensional vector space. Then $Gr(k,n)$ is a projective variety and it has Plucker coordinates $P_{i_1, ldots, i_k}$ ($i_1<ldots<i_k$) which is the determinant of the matrix $(x_{ij})_{i in [k], j in {i_1, ldots, i_k}}$. Certain Plucker coordinates satisfy the Plucker relation. For example, for $Gr(2,n)$, $P_{12}P_{34} + P_{23}P_{14}-P_{13}P_{24}=0$. Therefore $P_{13}P_{24} = P_{12}P_{34} + P_{23}P_{14}$ can be viewed as a decomposition of the product of $P_{13}$ and $P_{24}$. For $P_{12}$, $P_{34}$, we say that their product is irreducible. That is $P_{12}P_{34}$ cannot be written as a sum (with two or more terms in the summation, each summand has positive coefficient) of products of Plucker coordinates.
Given two Plucker coordinates $P_{i_1, ldots, i_k}$, $P_{j_1, ldots, j_k}$, is there some formula for the decomposition of $P_{i_1, ldots, i_k} P_{j_1, ldots, j_k} = sum_T c_T P_T$ (P_T is a product of certain Plucker coordinates, $c_T>0$) in the literature? Thank you very much.
ag.algebraic-geometry co.combinatorics
ag.algebraic-geometry co.combinatorics
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
asked Mar 30 at 12:57
Jianrong LiJianrong Li
2,53721319
2,53721319
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your exact question seems a little strange to me because, beyond the 3-term relation, more general Plücker relations will have many terms with both positive and negative signs. So for $k>2$ it is not clear that we can ever do decompositions of the type you're describing. But the question of which subsets of Plücker coordinates are algebraically independent and generate the coordinate ring of the Grassmannian, and how do we write arbitrary elements of the coordinate ring in the corresponding basis, is the beginning of the study of cluster algebras. See e.g. https://arxiv.org/abs/math/0311148.
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
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%2fmathoverflow.net%2fquestions%2f326748%2fdecomposition-of-product-of-two-plucker-coordinates%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your exact question seems a little strange to me because, beyond the 3-term relation, more general Plücker relations will have many terms with both positive and negative signs. So for $k>2$ it is not clear that we can ever do decompositions of the type you're describing. But the question of which subsets of Plücker coordinates are algebraically independent and generate the coordinate ring of the Grassmannian, and how do we write arbitrary elements of the coordinate ring in the corresponding basis, is the beginning of the study of cluster algebras. See e.g. https://arxiv.org/abs/math/0311148.
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
$endgroup$
add a comment |
$begingroup$
Your exact question seems a little strange to me because, beyond the 3-term relation, more general Plücker relations will have many terms with both positive and negative signs. So for $k>2$ it is not clear that we can ever do decompositions of the type you're describing. But the question of which subsets of Plücker coordinates are algebraically independent and generate the coordinate ring of the Grassmannian, and how do we write arbitrary elements of the coordinate ring in the corresponding basis, is the beginning of the study of cluster algebras. See e.g. https://arxiv.org/abs/math/0311148.
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
$endgroup$
add a comment |
$begingroup$
Your exact question seems a little strange to me because, beyond the 3-term relation, more general Plücker relations will have many terms with both positive and negative signs. So for $k>2$ it is not clear that we can ever do decompositions of the type you're describing. But the question of which subsets of Plücker coordinates are algebraically independent and generate the coordinate ring of the Grassmannian, and how do we write arbitrary elements of the coordinate ring in the corresponding basis, is the beginning of the study of cluster algebras. See e.g. https://arxiv.org/abs/math/0311148.
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
$endgroup$
Your exact question seems a little strange to me because, beyond the 3-term relation, more general Plücker relations will have many terms with both positive and negative signs. So for $k>2$ it is not clear that we can ever do decompositions of the type you're describing. But the question of which subsets of Plücker coordinates are algebraically independent and generate the coordinate ring of the Grassmannian, and how do we write arbitrary elements of the coordinate ring in the corresponding basis, is the beginning of the study of cluster algebras. See e.g. https://arxiv.org/abs/math/0311148.
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
answered Mar 30 at 15:23
Sam HopkinsSam Hopkins
5,24212560
5,24212560
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f326748%2fdecomposition-of-product-of-two-plucker-coordinates%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
