The Stabilizer of a Flag
More generally, the stabilizer of a flag (the linear operators on
\[V\]
such that \[T(V_i) \lt V_i\]
is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), where the block sizes is \[d_i-d_{i-1}\]
. The stabilizer subgroup of a complete flag is the set of invertible upper triangular matrices with respect to any basis adapted to the flag. The subgroup of lower triangular matrices with respect to such a basis depends on that basis, and can therefore not be characterized in terms of the flag only.The stabilizer subgroup of any complete flag is a Borel subgroup of the general linear group, and the stabilizer of any partial flags is a parabolic subgroup.
The stabilizer subgroup of a flag acts simply transitively on adapted bases for the flag, and thus these are not unique unless the stabilizer is trivial. That is a very exceptional circumstance: it happens only for a vector space of dimension 0, or for a vector space over
\[F_2\]
of dimension 1 - precisely the cases where only one basis exists, independently of any flag.
