An ascending chain of subspaces of a vector space. If 
is an 
-dimensional
vector space, a flag of 
is a filtration
 |
(1)
|
where all inclusions are strict. Hence
 |
(2)
|
so that 
.
If equality holds, then 
for all 
, and the flag is called complete or full. In this case it
is a composition series of 
.
A full flag can be constructed by fixing a basis 
of 
, and then taking 
for all 
.
A flag of any length can be obtained from a full flag by taking out some of the subspaces. Conversely, every flag can be completed to a full flag by inserting suitable subspaces.
In general, this can be done in different ways. The following flag of 
 |
(3)
|
can be completed by switching in any line of the 
-plane passing through the origin. Two different full flags
are, for example,
 |
(4)
|
and
 |
(5)
|
Schubert varieties are projective varieties defined from flags.
