<s>
In	O
mathematics	O
,	O
a	O
generalized	B-Algorithm
flag	I-Algorithm
variety	I-Algorithm
(	O
or	O
simply	O
flag	B-Algorithm
variety	I-Algorithm
)	O
is	O
a	O
homogeneous	O
space	O
whose	O
points	O
are	O
flags	O
in	O
a	O
finite-dimensional	O
vector	O
space	O
V	O
over	O
a	O
field	O
F	O
.	O
When	O
F	O
is	O
the	O
real	O
or	O
complex	O
numbers	O
,	O
a	O
generalized	B-Algorithm
flag	I-Algorithm
variety	I-Algorithm
is	O
a	O
smooth	O
or	O
complex	O
manifold	O
,	O
called	O
a	O
real	O
or	O
complex	O
flag	B-Algorithm
manifold	I-Algorithm
.	O
</s>
<s>
Flag	B-Algorithm
varieties	I-Algorithm
are	O
naturally	O
projective	O
varieties	O
.	O
</s>
<s>
Flag	B-Algorithm
varieties	I-Algorithm
can	O
be	O
defined	O
in	O
various	O
degrees	O
of	O
generality	O
.	O
</s>
<s>
A	O
prototype	O
is	O
the	O
variety	O
of	O
complete	O
flags	O
in	O
a	O
vector	O
space	O
V	O
over	O
a	O
field	O
F	O
,	O
which	O
is	O
a	O
flag	B-Algorithm
variety	I-Algorithm
for	O
the	O
special	O
linear	O
group	O
over	O
F	O
.	O
Other	O
flag	B-Algorithm
varieties	I-Algorithm
arise	O
by	O
considering	O
partial	O
flags	O
,	O
or	O
by	O
restriction	O
from	O
the	O
special	O
linear	O
group	O
to	O
subgroups	O
such	O
as	O
the	O
symplectic	O
group	O
.	O
</s>
<s>
In	O
the	O
most	O
general	O
sense	O
,	O
a	O
generalized	B-Algorithm
flag	I-Algorithm
variety	I-Algorithm
is	O
defined	O
to	O
mean	O
a	O
projective	B-Algorithm
homogeneous	I-Algorithm
variety	I-Algorithm
,	O
that	O
is	O
,	O
a	O
smooth	O
projective	O
variety	O
X	O
over	O
a	O
field	O
F	O
with	O
a	O
transitive	O
action	O
of	O
a	O
reductive	O
group	O
G	O
(	O
and	O
smooth	O
stabilizer	O
subgroup	O
;	O
that	O
is	O
no	O
restriction	O
for	O
F	O
of	O
characteristic	O
zero	O
)	O
.	O
</s>
<s>
If	O
X	O
has	O
an	O
F-rational	O
point	O
,	O
then	O
it	O
is	O
isomorphic	O
to	O
G/P	O
for	O
some	O
parabolic	O
subgroup	O
P	O
of	O
G	O
.	O
A	O
projective	B-Algorithm
homogeneous	I-Algorithm
variety	I-Algorithm
may	O
also	O
be	O
realised	O
as	O
the	O
orbit	O
of	O
a	O
highest	O
weight	O
vector	O
in	O
a	O
projectivized	O
representation	O
of	O
G	O
.	O
The	O
complex	O
projective	B-Algorithm
homogeneous	I-Algorithm
varieties	I-Algorithm
are	O
the	O
compact	O
flat	O
model	O
spaces	O
for	O
Cartan	O
geometries	O
of	O
parabolic	O
type	O
.	O
</s>
<s>
They	O
are	O
homogeneous	O
Riemannian	B-Architecture
manifolds	I-Architecture
under	O
any	O
maximal	O
compact	O
subgroup	O
of	O
G	O
,	O
and	O
they	O
are	O
precisely	O
the	O
coadjoint	O
orbits	O
of	O
compact	O
Lie	O
groups	O
.	O
</s>
<s>
Flag	B-Algorithm
manifolds	I-Algorithm
can	O
be	O
symmetric	O
spaces	O
.	O
</s>
<s>
Over	O
the	O
complex	O
numbers	O
,	O
the	O
corresponding	O
flag	B-Algorithm
manifolds	I-Algorithm
are	O
the	O
Hermitian	O
symmetric	O
spaces	O
.	O
</s>
<s>
Over	O
the	O
real	O
numbers	O
,	O
an	O
R-space	B-Algorithm
is	O
a	O
synonym	O
for	O
a	O
real	B-Algorithm
flag	I-Algorithm
manifold	I-Algorithm
and	O
the	O
corresponding	O
symmetric	O
spaces	O
are	O
called	O
symmetric	B-Algorithm
R-spaces	I-Algorithm
.	O
</s>
<s>
According	O
to	O
basic	O
results	O
of	O
linear	B-Language
algebra	I-Language
,	O
any	O
two	O
complete	O
flags	O
in	O
an	O
n-dimensional	O
vector	O
space	O
V	O
over	O
a	O
field	O
F	O
are	O
no	O
different	O
from	O
each	O
other	O
from	O
a	O
geometric	O
point	O
of	O
view	O
.	O
</s>
<s>
Relative	O
to	O
this	O
basis	O
,	O
the	O
stabilizer	O
of	O
the	O
standard	O
flag	O
is	O
the	O
group	O
of	O
nonsingular	O
lower	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
,	O
which	O
we	O
denote	O
by	O
Bn	O
.	O
</s>
<s>
The	O
complete	O
flag	B-Algorithm
variety	I-Algorithm
can	O
therefore	O
be	O
written	O
as	O
a	O
homogeneous	O
space	O
GL(n,F )	O
/	O
Bn	O
,	O
which	O
shows	O
in	O
particular	O
that	O
it	O
has	O
dimension	O
n(n1 )	O
/2	O
over	O
F	O
.	O
</s>
<s>
Note	O
that	O
the	O
multiples	O
of	O
the	O
identity	O
act	O
trivially	O
on	O
all	O
flags	O
,	O
and	O
so	O
one	O
can	O
restrict	O
attention	O
to	O
the	O
special	O
linear	O
group	O
SL(n,F )	O
of	O
matrices	O
with	O
determinant	O
one	O
,	O
which	O
is	O
a	O
semisimple	O
algebraic	O
group	O
;	O
the	O
set	O
of	O
lower	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
of	O
determinant	O
one	O
is	O
a	O
Borel	O
subgroup	O
.	O
</s>
<s>
If	O
the	O
field	O
F	O
is	O
the	O
real	O
or	O
complex	O
numbers	O
we	O
can	O
introduce	O
an	O
inner	O
product	O
on	O
V	O
such	O
that	O
the	O
chosen	O
basis	O
is	O
orthonormal	B-Algorithm
.	O
</s>
<s>
is	O
the	O
space	O
of	O
all	O
flags	O
of	O
signature	O
(	O
d1	O
,	O
d2	O
,	O
...	O
dk	O
)	O
in	O
a	O
vector	O
space	O
V	O
of	O
dimension	O
n	O
=	O
dk	O
over	O
F	O
.	O
The	O
complete	O
flag	B-Algorithm
variety	I-Algorithm
is	O
the	O
special	O
case	O
that	O
di	O
=	O
i	O
for	O
all	O
i	O
.	O
</s>
<s>
The	O
stabilizer	O
of	O
a	O
flag	O
of	O
nested	O
subspaces	O
Vi	O
of	O
dimension	O
di	O
can	O
be	O
taken	O
to	O
be	O
the	O
group	O
of	O
nonsingular	O
block	B-Algorithm
lower	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
,	O
where	O
the	O
dimensions	O
of	O
the	O
blocks	O
are	O
ni	O
:=	O
di	O
di−1	O
(	O
with	O
d0	O
=	O
0	O
)	O
.	O
</s>
<s>
Restricting	O
to	O
matrices	O
of	O
determinant	O
one	O
,	O
this	O
is	O
a	O
parabolic	O
subgroup	O
P	O
of	O
SL(n,F )	O
,	O
and	O
thus	O
the	O
partial	O
flag	B-Algorithm
variety	I-Algorithm
is	O
isomorphic	O
to	O
the	O
homogeneous	O
space	O
SL(n,F )	O
/P	O
.	O
</s>
<s>
The	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
of	O
determinant	O
one	O
are	O
a	O
Borel	O
subgroup	O
of	O
SL(n,F )	O
,	O
and	O
hence	O
the	O
stabilizers	O
of	O
partial	O
flags	O
are	O
parabolic	O
subgroups	O
.	O
</s>
<s>
Hence	O
,	O
more	O
generally	O
,	O
if	O
G	O
is	O
a	O
semisimple	O
algebraic	O
or	O
Lie	O
group	O
,	O
then	O
the	O
(	O
generalized	O
)	O
flag	B-Algorithm
variety	I-Algorithm
for	O
G	O
is	O
G/P	O
where	O
P	O
is	O
a	O
parabolic	O
subgroup	O
of	O
G	O
.	O
The	O
correspondence	O
between	O
parabolic	O
subgroups	O
and	O
generalized	B-Algorithm
flag	I-Algorithm
varieties	I-Algorithm
allows	O
each	O
to	O
be	O
understood	O
in	O
terms	O
of	O
the	O
other	O
.	O
</s>
<s>
The	O
extension	O
of	O
the	O
terminology	O
"	O
flag	B-Algorithm
variety	I-Algorithm
"	O
is	O
reasonable	O
,	O
because	O
points	O
of	O
G/P	O
can	O
still	O
be	O
described	O
using	O
flags	O
.	O
</s>
<s>
This	O
variety	O
is	O
a	O
(	O
generalized	O
)	O
flag	B-Algorithm
variety	I-Algorithm
,	O
and	O
furthermore	O
,	O
every	O
(	O
generalized	O
)	O
flag	B-Algorithm
variety	I-Algorithm
for	O
G	O
arises	O
in	O
this	O
way	O
.	O
</s>
<s>
Armand	O
Borel	O
showed	O
that	O
this	O
characterizes	O
the	O
flag	B-Algorithm
varieties	I-Algorithm
of	O
a	O
general	O
semisimple	O
algebraic	O
group	O
G	O
:	O
they	O
are	O
precisely	O
the	O
complete	O
homogeneous	O
spaces	O
of	O
G	O
,	O
or	O
equivalently	O
(	O
in	O
this	O
context	O
)	O
,	O
the	O
projective	O
homogeneous	O
G-varieties	O
.	O
</s>
<s>
Let	O
G	O
be	O
a	O
semisimple	O
Lie	O
group	O
with	O
maximal	O
compact	O
subgroup	O
K	O
.	O
Then	O
K	O
acts	O
transitively	O
on	O
any	O
conjugacy	O
class	O
of	O
parabolic	O
subgroups	O
,	O
and	O
hence	O
the	O
generalized	B-Algorithm
flag	I-Algorithm
variety	I-Algorithm
G/P	O
is	O
a	O
compact	O
homogeneous	O
Riemannian	B-Architecture
manifold	I-Architecture
K/	O
( KP	O
)	O
with	O
isometry	O
group	O
K	O
.	O
Furthermore	O
,	O
if	O
G	O
is	O
a	O
complex	O
Lie	O
group	O
,	O
G/P	O
is	O
a	O
homogeneous	O
Kähler	O
manifold	O
.	O
</s>
<s>
Over	O
the	O
real	O
numbers	O
,	O
a	O
real	B-Algorithm
flag	I-Algorithm
manifold	I-Algorithm
is	O
also	O
called	O
an	O
R-space	B-Algorithm
,	O
and	O
the	O
R-spaces	B-Algorithm
which	O
are	O
Riemannian	O
symmetric	O
spaces	O
under	O
K	O
are	O
known	O
as	O
symmetric	B-Algorithm
R-spaces	I-Algorithm
.	O
</s>
<s>
The	O
symmetric	B-Algorithm
R-spaces	I-Algorithm
which	O
are	O
not	O
Hermitian	O
symmetric	O
are	O
obtained	O
by	O
taking	O
G	O
to	O
be	O
a	O
real	O
form	O
of	O
the	O
biholomorphism	O
group	O
Gc	O
of	O
a	O
Hermitian	O
symmetric	O
space	O
Gc/Pc	O
such	O
that	O
P	O
:=	O
PcG	O
is	O
a	O
parabolic	O
subgroup	O
of	O
G	O
.	O
Examples	O
include	O
projective	O
spaces	O
(	O
with	O
G	O
the	O
group	O
of	O
projective	B-Algorithm
transformations	I-Algorithm
)	O
and	O
spheres	O
(	O
with	O
G	O
the	O
group	O
of	O
conformal	O
transformations	O
)	O
.	O
</s>
