# Flag

*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 $ 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).

For references see Flag structure.

**How to Cite This Entry:**

Flag.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Flag&oldid=44248