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''of type $  \nu $ 
in an $  n $ -
dimensional vector space $  \nu $ ''


A collection of linear subspaces $  V $ 
of $  V _{1} \dots V _{k} $ 
of corresponding dimensions $  V $ , 
such that $  n _{1} \dots n _{k} $ (
here $  V _{1} \subset \dots \subset V _{k} $ , 
$  \nu = (n _{1} \dots n _{k} ) $ ; 
$  1 \leq k \leq n - 1 ;\quad 0 < n _{1} < \dots < n _{k} < n $ ). 
A flag of type $  \nu _{0} = (1 \dots n - 1 ) $ 
is called a complete flag or a full flag. Any two flags of the same type can be mapped to each other by some linear transformation of $  V $ , 
that is, the set $  F _ \nu  (V) $ 
of all flags of type $  \nu $ 
in $  V $ 
is a homogeneous space of the general linear group $   \mathop{\rm GL}\nolimits (V) $ . 
The unimodular group $   \mathop{\rm SL}\nolimits (V) $ 
also acts transitively on $  F _ \nu  (V) $ . 
Here the stationary subgroup $  H _{F} $ 
of $  F $ 
in $   \mathop{\rm GL}\nolimits (V) $ (
and also in $   \mathop{\rm SL}\nolimits (V) $ ) 
is a parabolic subgroup of $   \mathop{\rm GL}\nolimits (V) $ (
respectively, of $   \mathop{\rm SL}\nolimits (V) $ ). 
If $  F $ 
is a complete flag in $  V $ , 
defined by subspaces $  V _{1} \subset \dots \subset V _ {n - 1} $ , 
then $  H _{F} $ 
is a complete triangular subgroup of $   \mathop{\rm GL}\nolimits (V) $ (
respectively, of $   \mathop{\rm SL}\nolimits (V) $ ) 
relative to a basis $  e _{1} \dots e _{n} $ 
of $  V $ 
such that $  e _{i} \in V _{i} $ , 
$  i = 1 \dots n $ . 
In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For $  k = 1 $ , 
a flag of type $  (n _{1} ) $ 
is simply an $  n _{1} $ -
dimensional linear subspace of $  V $ 
and $  F _ {(n _{1} )} (V) $ 
is the [[Grassmann manifold|Grassmann manifold]] $  G _ {n, n _{1}} =  \mathop{\rm Gr}\nolimits _ {n _{1}} (V) $ . 
In particular, $  F _{(1)} (V) $ 
is the projective space associated with the vector space $  V $ . 
Every flag variety $  F _ \nu  (V) $ 
can be canonically equipped with the structure of a projective algebraic variety (see ). If $  V $ 
is a real or complex vector space, then all the varieties $  F _ \nu  (V) $ 
are compact. Cellular decompositions and cohomology rings of the $  F _ \nu  (V) $ 
are known (see , and also [[Bruhat decomposition|Bruhat decomposition]]).

For references see [[Flag structure|Flag structure]].