Plucker relations in orthogonal GrassmannianAction on the highest weight vector of a representation of a semisimple linear algebraic groupCriterion for nilradical of a maximal parabolic subalgebra to be abelian?Grassmann-Plücker relations for permanentsAre Strata of the affine Grassmannian total spaces of equivariant vector bundles over flag varietiesVarious definitions of the Bruhat decomposition of the affine GrassmannianIntegral lattices in Lie group representationsToric variety defined by the Weyl orbit of a minuscule weightCounting cosets in the Quotient of Weyl groupsConcrete description of an exceptional minuscule varietyPlucker coordinates of flag varieties
Plucker relations in orthogonal Grassmannian
Action on the highest weight vector of a representation of a semisimple linear algebraic groupCriterion for nilradical of a maximal parabolic subalgebra to be abelian?Grassmann-Plücker relations for permanentsAre Strata of the affine Grassmannian total spaces of equivariant vector bundles over flag varietiesVarious definitions of the Bruhat decomposition of the affine GrassmannianIntegral lattices in Lie group representationsToric variety defined by the Weyl orbit of a minuscule weightCounting cosets in the Quotient of Weyl groupsConcrete description of an exceptional minuscule varietyPlucker coordinates of flag varieties
$begingroup$
Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $varpi_3$. Since $varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known in literature ? What are the Plücker type relations (quadratic) in this case ? Are the relations same as they are in the Grassmannian $G(3,7)$ ? I have a feeling that the quotient is $mathbb P^2$.
ag.algebraic-geometry rt.representation-theory algebraic-groups algebraic-combinatorics
edited May 15 at 13:35
Glorfindel
1,35441221
asked May 15 at 1:46
icmes imrficmes imrf
613
$endgroup$
add a comment |
$begingroup$
Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $varpi_3$. Since $varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known in literature ? What are the Plücker type relations (quadratic) in this case ? Are the relations same as they are in the Grassmannian $G(3,7)$ ? I have a feeling that the quotient is $mathbb P^2$.
ag.algebraic-geometry rt.representation-theory algebraic-groups algebraic-combinatorics
edited May 15 at 13:35
Glorfindel
1,35441221
asked May 15 at 1:46
icmes imrficmes imrf
613
$endgroup$
add a comment |
$begingroup$
Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $varpi_3$. Since $varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known in literature ? What are the Plücker type relations (quadratic) in this case ? Are the relations same as they are in the Grassmannian $G(3,7)$ ? I have a feeling that the quotient is $mathbb P^2$.
ag.algebraic-geometry rt.representation-theory algebraic-groups algebraic-combinatorics
edited May 15 at 13:35
Glorfindel
1,35441221
asked May 15 at 1:46
icmes imrficmes imrf
613
$endgroup$
Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $varpi_3$. Since $varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known in literature ? What are the Plücker type relations (quadratic) in this case ? Are the relations same as they are in the Grassmannian $G(3,7)$ ? I have a feeling that the quotient is $mathbb P^2$.
ag.algebraic-geometry rt.representation-theory algebraic-groups algebraic-combinatorics
ag.algebraic-geometry rt.representation-theory algebraic-groups algebraic-combinatorics
edited May 15 at 13:35
Glorfindel
1,35441221
asked May 15 at 1:46
icmes imrficmes imrf
613
edited May 15 at 13:35
Glorfindel
1,35441221
asked May 15 at 1:46
icmes imrficmes imrf
613
edited May 15 at 13:35
Glorfindel
1,35441221
edited May 15 at 13:35
Glorfindel
1,35441221
edited May 15 at 13:35
Glorfindel
1,35441221
1,35441221
asked May 15 at 1:46
icmes imrficmes imrf
613
asked May 15 at 1:46
icmes imrficmes imrf
613
asked May 15 at 1:46
icmes imrficmes imrf
613
613
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
If $G = SO(7)$ and $P$ corresponds to $varpi_3$, then
$$
G/P cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).
The isomorphism can be sen as a combination of a general isomorphism
$$
SO(2n-1)/P_varpi_n-1 cong SO(2n)/P_varpi_n
$$
and the triality isomorphism
$$
SO(8)/P_varpi_4 cong SO(8)/P_varpi_1 cong Q^6.
$$
edited May 15 at 13:35
Glorfindel
1,35441221
answered May 15 at 4:23
SashaSasha
21.5k22857
$endgroup$
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
add a comment |
$begingroup$
Standard Monomial Theory (https://en.wikipedia.org/wiki/Standard_monomial_theory) provides a uniform description of the coordinate ring of any minuscule $G/P$ (and in fact with more work applies to other $G/P$ as well). In the minuscule case the main result, due to Seshadri, is that a basis of the coordinate ring is given by multichains in the corresponding minuscule poset. However, this result only says that in principle there is some way to express a product of standard monomials as a sum of standard monomials (in other words, we have an "algebra with a straightening law"). But we might not know the explicit coefficients in this expression.
Fortunately, for the case you care about (the maximal orthogonal Grassmannian) the specific relations actually have been worked out. Namely, the paper "Pfaffians and Shuffling Relations for the Spin Module" (https://arxiv.org/abs/1203.2943) by Chirivì and Maffei contains the exact relations (phrased in terms of pfaffians of skew-symmetric matrices) you're looking for.
There is one small complication in that Chirivì and Maffei work with $So(2n+2)$ (Type D) and you're interested in $So(2n+1)$ (Type B), but if I remember correctly we actually have an isomorphism of the relevant varieties $OG(n+1,2n+2)simeq OG(n,2n+1)$, so this makes no difference.
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
$endgroup$
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
1
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
add a comment |
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2 Answers
2
active
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votes
2 Answers
2
active
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$begingroup$
If $G = SO(7)$ and $P$ corresponds to $varpi_3$, then
$$
G/P cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).
The isomorphism can be sen as a combination of a general isomorphism
$$
SO(2n-1)/P_varpi_n-1 cong SO(2n)/P_varpi_n
$$
and the triality isomorphism
$$
SO(8)/P_varpi_4 cong SO(8)/P_varpi_1 cong Q^6.
$$
edited May 15 at 13:35
Glorfindel
1,35441221
answered May 15 at 4:23
SashaSasha
21.5k22857
$endgroup$
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
add a comment |
$begingroup$
If $G = SO(7)$ and $P$ corresponds to $varpi_3$, then
$$
G/P cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).
The isomorphism can be sen as a combination of a general isomorphism
$$
SO(2n-1)/P_varpi_n-1 cong SO(2n)/P_varpi_n
$$
and the triality isomorphism
$$
SO(8)/P_varpi_4 cong SO(8)/P_varpi_1 cong Q^6.
$$
edited May 15 at 13:35
Glorfindel
1,35441221
answered May 15 at 4:23
SashaSasha
21.5k22857
$endgroup$
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
add a comment |
$begingroup$
If $G = SO(7)$ and $P$ corresponds to $varpi_3$, then
$$
G/P cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).
The isomorphism can be sen as a combination of a general isomorphism
$$
SO(2n-1)/P_varpi_n-1 cong SO(2n)/P_varpi_n
$$
and the triality isomorphism
$$
SO(8)/P_varpi_4 cong SO(8)/P_varpi_1 cong Q^6.
$$
edited May 15 at 13:35
Glorfindel
1,35441221
answered May 15 at 4:23
SashaSasha
21.5k22857
$endgroup$
If $G = SO(7)$ and $P$ corresponds to $varpi_3$, then
$$
G/P cong Q^6
$$
is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).
The isomorphism can be sen as a combination of a general isomorphism
$$
SO(2n-1)/P_varpi_n-1 cong SO(2n)/P_varpi_n
$$
and the triality isomorphism
$$
SO(8)/P_varpi_4 cong SO(8)/P_varpi_1 cong Q^6.
$$
edited May 15 at 13:35
Glorfindel
1,35441221
answered May 15 at 4:23
SashaSasha
21.5k22857
edited May 15 at 13:35
Glorfindel
1,35441221
edited May 15 at 13:35
Glorfindel
1,35441221
edited May 15 at 13:35
Glorfindel
1,35441221
1,35441221
answered May 15 at 4:23
SashaSasha
21.5k22857
answered May 15 at 4:23
SashaSasha
21.5k22857
answered May 15 at 4:23
SashaSasha
21.5k22857
21.5k22857
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
add a comment |
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Ah, good point about triality!
$endgroup$
– Sam Hopkins
May 15 at 4:32
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
$begingroup$
Is there a reference why $SO(8)/P_varpi_1 cong Q^6$ ?
$endgroup$
– icmes imrf
May 16 at 0:56
add a comment |
$begingroup$
Standard Monomial Theory (https://en.wikipedia.org/wiki/Standard_monomial_theory) provides a uniform description of the coordinate ring of any minuscule $G/P$ (and in fact with more work applies to other $G/P$ as well). In the minuscule case the main result, due to Seshadri, is that a basis of the coordinate ring is given by multichains in the corresponding minuscule poset. However, this result only says that in principle there is some way to express a product of standard monomials as a sum of standard monomials (in other words, we have an "algebra with a straightening law"). But we might not know the explicit coefficients in this expression.
Fortunately, for the case you care about (the maximal orthogonal Grassmannian) the specific relations actually have been worked out. Namely, the paper "Pfaffians and Shuffling Relations for the Spin Module" (https://arxiv.org/abs/1203.2943) by Chirivì and Maffei contains the exact relations (phrased in terms of pfaffians of skew-symmetric matrices) you're looking for.
There is one small complication in that Chirivì and Maffei work with $So(2n+2)$ (Type D) and you're interested in $So(2n+1)$ (Type B), but if I remember correctly we actually have an isomorphism of the relevant varieties $OG(n+1,2n+2)simeq OG(n,2n+1)$, so this makes no difference.
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
$endgroup$
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
1
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
add a comment |
$begingroup$
Standard Monomial Theory (https://en.wikipedia.org/wiki/Standard_monomial_theory) provides a uniform description of the coordinate ring of any minuscule $G/P$ (and in fact with more work applies to other $G/P$ as well). In the minuscule case the main result, due to Seshadri, is that a basis of the coordinate ring is given by multichains in the corresponding minuscule poset. However, this result only says that in principle there is some way to express a product of standard monomials as a sum of standard monomials (in other words, we have an "algebra with a straightening law"). But we might not know the explicit coefficients in this expression.
Fortunately, for the case you care about (the maximal orthogonal Grassmannian) the specific relations actually have been worked out. Namely, the paper "Pfaffians and Shuffling Relations for the Spin Module" (https://arxiv.org/abs/1203.2943) by Chirivì and Maffei contains the exact relations (phrased in terms of pfaffians of skew-symmetric matrices) you're looking for.
There is one small complication in that Chirivì and Maffei work with $So(2n+2)$ (Type D) and you're interested in $So(2n+1)$ (Type B), but if I remember correctly we actually have an isomorphism of the relevant varieties $OG(n+1,2n+2)simeq OG(n,2n+1)$, so this makes no difference.
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
$endgroup$
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
1
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
add a comment |
$begingroup$
Standard Monomial Theory (https://en.wikipedia.org/wiki/Standard_monomial_theory) provides a uniform description of the coordinate ring of any minuscule $G/P$ (and in fact with more work applies to other $G/P$ as well). In the minuscule case the main result, due to Seshadri, is that a basis of the coordinate ring is given by multichains in the corresponding minuscule poset. However, this result only says that in principle there is some way to express a product of standard monomials as a sum of standard monomials (in other words, we have an "algebra with a straightening law"). But we might not know the explicit coefficients in this expression.
Fortunately, for the case you care about (the maximal orthogonal Grassmannian) the specific relations actually have been worked out. Namely, the paper "Pfaffians and Shuffling Relations for the Spin Module" (https://arxiv.org/abs/1203.2943) by Chirivì and Maffei contains the exact relations (phrased in terms of pfaffians of skew-symmetric matrices) you're looking for.
There is one small complication in that Chirivì and Maffei work with $So(2n+2)$ (Type D) and you're interested in $So(2n+1)$ (Type B), but if I remember correctly we actually have an isomorphism of the relevant varieties $OG(n+1,2n+2)simeq OG(n,2n+1)$, so this makes no difference.
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
$endgroup$
Standard Monomial Theory (https://en.wikipedia.org/wiki/Standard_monomial_theory) provides a uniform description of the coordinate ring of any minuscule $G/P$ (and in fact with more work applies to other $G/P$ as well). In the minuscule case the main result, due to Seshadri, is that a basis of the coordinate ring is given by multichains in the corresponding minuscule poset. However, this result only says that in principle there is some way to express a product of standard monomials as a sum of standard monomials (in other words, we have an "algebra with a straightening law"). But we might not know the explicit coefficients in this expression.
Fortunately, for the case you care about (the maximal orthogonal Grassmannian) the specific relations actually have been worked out. Namely, the paper "Pfaffians and Shuffling Relations for the Spin Module" (https://arxiv.org/abs/1203.2943) by Chirivì and Maffei contains the exact relations (phrased in terms of pfaffians of skew-symmetric matrices) you're looking for.
There is one small complication in that Chirivì and Maffei work with $So(2n+2)$ (Type D) and you're interested in $So(2n+1)$ (Type B), but if I remember correctly we actually have an isomorphism of the relevant varieties $OG(n+1,2n+2)simeq OG(n,2n+1)$, so this makes no difference.
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
edited May 15 at 13:46
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
answered May 15 at 1:49
Sam HopkinsSam Hopkins
5,73212663
5,73212663
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
1
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
add a comment |
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
1
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
$begingroup$
Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces.
$endgroup$
– icmes imrf
May 15 at 2:06
1
1
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
$begingroup$
@icmesimrf: see my edited answer for a paper with the exact relations you're looking for.
$endgroup$
– Sam Hopkins
May 15 at 2:23
add a comment |
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