\[V\]
.\[\mathbf{0} \subset V_1 \subset V_2 \subset ... \subset V_n =V\]
If the dimension of
\[V_i\]
is \[d_i\]
then \[d_0 \lt d_1 \lt d_2 \lt d_3 \lt ...\lt d_k =n\]
where
\[n\]
is the dimension of \[V\]
. A flag is called a complete flag if \[d_i =i\]
, otherwise it is called a partial flag. The sequence \[d_1, \: d_2 , \: d_3, \: d_4 \: ... \: d_k\]
is called the signature of the flag.