Flag
of type 
in an 
-dimensional vector space 
A collection of linear subspaces 
of 
of corresponding dimensions 
, such that 
(here 
, 
; 
). A flag of type 
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 
, that is, the set 
of all flags of type 
in 
is a homogeneous space of the general linear group 
. The unimodular group 
also acts transitively on 
. Here the stationary subgroup 
of 
in 
(and also in 
) is a parabolic subgroup of 
(respectively, of 
). If 
is a complete flag in 
, defined by subspaces 
, then 
is a complete triangular subgroup of 
(respectively, of 
) relative to a basis 
of 
such that 
, 
. In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For 
, a flag of type 
is simply an 
-dimensional linear subspace of 
and 
is the Grassmann manifold 
. In particular, 
is the projective space associated with the vector space 
. Every flag variety 
can be canonically equipped with the structure of a projective algebraic variety (see [1]). If 
is a real or complex vector space, then all the varieties 
are compact. Cellular decompositions and cohomology rings of the 
are known (see [3], and also Bruhat decomposition).
For references see Flag structure.
Flag. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flag&oldid=44248