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=19141