Group $ GL(V) $ acts naturally on $ mathscrF $, how to get its orbits? The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)the group acts faithfully on the linecomputing the orbits for a group actionDifference between “$G$ acts on $A$” and “G is a permutation group on $A$ (i.e. $Gleq S_A$)”Finding the orbits of the orthogonal group $O(n)$ on $Bbb R^n$Group of deck transformations acts properly discontinuouslyNon-trivial blocks when $S_n$ acts naturally on $[n]$.An action of $mathbbZ_n$ on complex $S^1$ and its geometric meaningCan a group action be defined by its orbits?Let H be a subgroup of finite group G. G acts on G/H by left multiplication. This induces a homomorphism. Show that its kernel is in HGroup actions and orbits
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Group $ GL(V) $ acts naturally on $ mathscrF $, how to get its orbits?
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)the group acts faithfully on the linecomputing the orbits for a group actionDifference between “$G$ acts on $A$” and “G is a permutation group on $A$ (i.e. $Gleq S_A$)”Finding the orbits of the orthogonal group $O(n)$ on $Bbb R^n$Group of deck transformations acts properly discontinuouslyNon-trivial blocks when $S_n$ acts naturally on $[n]$.An action of $mathbbZ_n$ on complex $S^1$ and its geometric meaningCan a group action be defined by its orbits?Let H be a subgroup of finite group G. G acts on G/H by left multiplication. This induces a homomorphism. Show that its kernel is in HGroup actions and orbits
$begingroup$
Could you please explain to me the meaning of the marking? Thanks in advance.
group-actions
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
$endgroup$
add a comment |
$begingroup$
Could you please explain to me the meaning of the marking? Thanks in advance.
group-actions
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
$endgroup$
add a comment |
$begingroup$
Could you please explain to me the meaning of the marking? Thanks in advance.
group-actions
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
$endgroup$
Could you please explain to me the meaning of the marking? Thanks in advance.
group-actions
group-actions
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
asked Mar 31 at 8:25
FyhswdsxjjFyhswdsxjj
796
796
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
This is basics of flag varieties.
$GL(V)$ acts transitively on each set of flags $V_1 subset dots subset V_n$ where the dimensions of the subspaces $V_i$ are fixed (this is basic linear algebra).
Equivalently, we can describe the dimensions of the subspaces $V_i$ by giving $dim V_1$ and the dimensions of the quotient vector spaces $V_i+1/V_i$ (which is simply $dim V_i+1 - dim V_i$.
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
$endgroup$
add a comment |
$begingroup$
What it is saying is that if you have two flags
$$
f=(V_1subsetcdotssubset V_n)quadtextandquad g=(W_1subsetcdotssubset W_n)
$$
in $cal F$, you can find $ginrm GL(V)$ such that
$$
g(V_i)=W_iquadtextfor allquad i=1,...,n
$$
if and only if
$$
dim(V_i+1/V_i)=dim(W_i+1/W_i)quadtextfor allquad i=1,...,n.
$$
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
$endgroup$
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
|
show 1 more comment
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
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oldest
votes
$begingroup$
This is basics of flag varieties.
$GL(V)$ acts transitively on each set of flags $V_1 subset dots subset V_n$ where the dimensions of the subspaces $V_i$ are fixed (this is basic linear algebra).
Equivalently, we can describe the dimensions of the subspaces $V_i$ by giving $dim V_1$ and the dimensions of the quotient vector spaces $V_i+1/V_i$ (which is simply $dim V_i+1 - dim V_i$.
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
$endgroup$
add a comment |
$begingroup$
This is basics of flag varieties.
$GL(V)$ acts transitively on each set of flags $V_1 subset dots subset V_n$ where the dimensions of the subspaces $V_i$ are fixed (this is basic linear algebra).
Equivalently, we can describe the dimensions of the subspaces $V_i$ by giving $dim V_1$ and the dimensions of the quotient vector spaces $V_i+1/V_i$ (which is simply $dim V_i+1 - dim V_i$.
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
$endgroup$
add a comment |
$begingroup$
This is basics of flag varieties.
$GL(V)$ acts transitively on each set of flags $V_1 subset dots subset V_n$ where the dimensions of the subspaces $V_i$ are fixed (this is basic linear algebra).
Equivalently, we can describe the dimensions of the subspaces $V_i$ by giving $dim V_1$ and the dimensions of the quotient vector spaces $V_i+1/V_i$ (which is simply $dim V_i+1 - dim V_i$.
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
$endgroup$
This is basics of flag varieties.
$GL(V)$ acts transitively on each set of flags $V_1 subset dots subset V_n$ where the dimensions of the subspaces $V_i$ are fixed (this is basic linear algebra).
Equivalently, we can describe the dimensions of the subspaces $V_i$ by giving $dim V_1$ and the dimensions of the quotient vector spaces $V_i+1/V_i$ (which is simply $dim V_i+1 - dim V_i$.
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
answered Mar 31 at 8:31
AnonymousCowardAnonymousCoward
3,1422436
3,1422436
add a comment |
add a comment |
$begingroup$
What it is saying is that if you have two flags
$$
f=(V_1subsetcdotssubset V_n)quadtextandquad g=(W_1subsetcdotssubset W_n)
$$
in $cal F$, you can find $ginrm GL(V)$ such that
$$
g(V_i)=W_iquadtextfor allquad i=1,...,n
$$
if and only if
$$
dim(V_i+1/V_i)=dim(W_i+1/W_i)quadtextfor allquad i=1,...,n.
$$
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
$endgroup$
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
|
show 1 more comment
$begingroup$
What it is saying is that if you have two flags
$$
f=(V_1subsetcdotssubset V_n)quadtextandquad g=(W_1subsetcdotssubset W_n)
$$
in $cal F$, you can find $ginrm GL(V)$ such that
$$
g(V_i)=W_iquadtextfor allquad i=1,...,n
$$
if and only if
$$
dim(V_i+1/V_i)=dim(W_i+1/W_i)quadtextfor allquad i=1,...,n.
$$
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
$endgroup$
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
|
show 1 more comment
$begingroup$
What it is saying is that if you have two flags
$$
f=(V_1subsetcdotssubset V_n)quadtextandquad g=(W_1subsetcdotssubset W_n)
$$
in $cal F$, you can find $ginrm GL(V)$ such that
$$
g(V_i)=W_iquadtextfor allquad i=1,...,n
$$
if and only if
$$
dim(V_i+1/V_i)=dim(W_i+1/W_i)quadtextfor allquad i=1,...,n.
$$
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
$endgroup$
What it is saying is that if you have two flags
$$
f=(V_1subsetcdotssubset V_n)quadtextandquad g=(W_1subsetcdotssubset W_n)
$$
in $cal F$, you can find $ginrm GL(V)$ such that
$$
g(V_i)=W_iquadtextfor allquad i=1,...,n
$$
if and only if
$$
dim(V_i+1/V_i)=dim(W_i+1/W_i)quadtextfor allquad i=1,...,n.
$$
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
answered Mar 31 at 8:35
Andrea MoriAndrea Mori
20.2k13466
20.2k13466
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
|
show 1 more comment
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
Of course, you also need to know $dim V_1$ (or one of the other dimensions) for this statement to be true.
$endgroup$
– AnonymousCoward
Mar 31 at 8:42
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
@AnonymousCoward: since $dim V_n=dim W_n=dim V=d$ that comes for free, doesn't it?
$endgroup$
– Andrea Mori
Mar 31 at 8:47
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Thanks! If I replace $mathscrF$ with $mathscrSF$(super flag varieties), $GL(V)$ with $GL(p|q)$, what will it become?
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:18
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
Sorry, it means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:25
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
$begingroup$
"it" means $Theta_d$($GL(p|q)/mathscrSFtimes mathscrSF$).
$endgroup$
– Fyhswdsxjj
Mar 31 at 9:26
|
show 1 more comment
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