Editing Flag (linear algebra) (section)

==Stabilizer==
The stabilizer subgroup of the standard flag is the [[group (mathematics)|group]] of [[invertible matrix|invertible]] [[upper triangular]] [[matrix (mathematics)|matrices]].

More generally, the stabilizer of a flag (the [[linear operators]] <math>T</math> on ''V'' such that <math>T(V_i) < V_i</math> for all ''i'') is, in matrix terms, the [[algebra over a field|algebra]] of block [[upper triangular]] matrices (with respect to an adapted basis), where the block sizes are <math>d_i-d_{i-1}</math>. 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 [[Borel subgroup#Parabolic subgroups|parabolic subgroup]].

The stabilizer subgroup of a flag acts [[simply transitive]]ly 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 <math>\mathbf{F}_2</math> of dimension 1 (precisely the cases where only one basis exists, independently of any flag).