Projective spaces and flag manifolds
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Is the complex projective space $mathbb CP^2$ a flag variety? If yes, what are the complex semisimple Lie group $S$ and a parabolic subgroup $H$ such that $mathbb CP^2=S/H$?
differential-geometry algebraic-geometry lie-groups
asked Nov 22 at 18:37
Amrat A
31818
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up vote
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Is the complex projective space $mathbb CP^2$ a flag variety? If yes, what are the complex semisimple Lie group $S$ and a parabolic subgroup $H$ such that $mathbb CP^2=S/H$?
differential-geometry algebraic-geometry lie-groups
asked Nov 22 at 18:37
Amrat A
31818
2
The group is $SL_3$, and any maximal parabolic in it.
– Sasha
Nov 22 at 18:52
@Sasha So is it true in general that all projective spaces are flag manifolds?
– Amrat A
Nov 22 at 19:03
2
Yes, it is true in general.
– Sasha
Nov 22 at 19:12
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Is the complex projective space $mathbb CP^2$ a flag variety? If yes, what are the complex semisimple Lie group $S$ and a parabolic subgroup $H$ such that $mathbb CP^2=S/H$?
differential-geometry algebraic-geometry lie-groups
asked Nov 22 at 18:37
Amrat A
31818
Is the complex projective space $mathbb CP^2$ a flag variety? If yes, what are the complex semisimple Lie group $S$ and a parabolic subgroup $H$ such that $mathbb CP^2=S/H$?
differential-geometry algebraic-geometry lie-groups
differential-geometry algebraic-geometry lie-groups
asked Nov 22 at 18:37
Amrat A
31818
asked Nov 22 at 18:37
Amrat A
31818
asked Nov 22 at 18:37
Amrat A
31818
asked Nov 22 at 18:37
Amrat A
31818
asked Nov 22 at 18:37
Amrat A
31818
31818
2
The group is $SL_3$, and any maximal parabolic in it.
– Sasha
Nov 22 at 18:52
@Sasha So is it true in general that all projective spaces are flag manifolds?
– Amrat A
Nov 22 at 19:03
2
Yes, it is true in general.
– Sasha
Nov 22 at 19:12
add a comment |
2
The group is $SL_3$, and any maximal parabolic in it.
– Sasha
Nov 22 at 18:52
@Sasha So is it true in general that all projective spaces are flag manifolds?
– Amrat A
Nov 22 at 19:03
2
Yes, it is true in general.
– Sasha
Nov 22 at 19:12
2
2
The group is $SL_3$, and any maximal parabolic in it.
– Sasha
Nov 22 at 18:52
The group is $SL_3$, and any maximal parabolic in it.
– Sasha
Nov 22 at 18:52
@Sasha So is it true in general that all projective spaces are flag manifolds?
– Amrat A
Nov 22 at 19:03
@Sasha So is it true in general that all projective spaces are flag manifolds?
– Amrat A
Nov 22 at 19:03
2
2
Yes, it is true in general.
– Sasha
Nov 22 at 19:12
Yes, it is true in general.
– Sasha
Nov 22 at 19:12
add a comment |
1 Answer
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More generally, for any sequence $d_1 < d_2 < dots < d_k < n$, the space of all partial flags $$V_1 subset V_2 subset dots subset V_k subset Bbb C^n$$
with $dim V_i = d_i$ has a transitive $SL_n(Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
More generally, for any sequence $d_1 < d_2 < dots < d_k < n$, the space of all partial flags $$V_1 subset V_2 subset dots subset V_k subset Bbb C^n$$
with $dim V_i = d_i$ has a transitive $SL_n(Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
add a comment |
up vote
0
down vote
accepted
More generally, for any sequence $d_1 < d_2 < dots < d_k < n$, the space of all partial flags $$V_1 subset V_2 subset dots subset V_k subset Bbb C^n$$
with $dim V_i = d_i$ has a transitive $SL_n(Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
More generally, for any sequence $d_1 < d_2 < dots < d_k < n$, the space of all partial flags $$V_1 subset V_2 subset dots subset V_k subset Bbb C^n$$
with $dim V_i = d_i$ has a transitive $SL_n(Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
More generally, for any sequence $d_1 < d_2 < dots < d_k < n$, the space of all partial flags $$V_1 subset V_2 subset dots subset V_k subset Bbb C^n$$
with $dim V_i = d_i$ has a transitive $SL_n(Bbb C)$-action. You can easily write down the stabiliser and more or less by definition it's a parabolic subgroup. Your question is the special case when $n=3, k=1$ and $d_1 = 1$.
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
answered Nov 23 at 10:53
Nicolas Hemelsoet
5,7352417
5,7352417
add a comment |
add a comment |
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2
The group is $SL_3$, and any maximal parabolic in it.
– Sasha
Nov 22 at 18:52
@Sasha So is it true in general that all projective spaces are flag manifolds?
– Amrat A
Nov 22 at 19:03
2
Yes, it is true in general.
– Sasha
Nov 22 at 19:12