In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector spaceV. Here "increasing" means each is a proper subspace of the next : If we write the dim Vi = di then we have where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if di = i for all i, otherwise it is called a partial flag. A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed by inserting suitable subspaces. The signature of the flag is the sequence. Under certain conditions the resulting sequence resembles a flag with a point connected to a line connected to a surface.
Bases
An ordered basis for V is said to be adapted to a flag if the first dibasis vectors form a basis for Vi for each 0 ≤ i ≤ k. Standard arguments from linear algebra can show that any flag has an adapted basis. Any ordered basis gives rise to a complete flag by letting the Vi be the span of the first i basis vectors. For example, the in Rn is induced from the standard basis where ei denotes the vector with a 1 in the ith slot and 0's elsewhere. Concretely, the standard flag is the subspaces: An adapted basis is almost never unique ; see below. A complete flag on an inner product space has an essentially unique orthonormal basis: it is unique up to multiplying each vector by a unit. This is easiest to prove inductively, by noting that, which defines it uniquely up to unit. More abstractly, it is unique up to an action of the maximal torus: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup.
Stabilizer
The stabilizer subgroup of the standard flag is the group of invertible upper triangular matrices. More generally, the stabilizer of a flag is, in matrix terms, the algebra of block upper triangular matrices, where the block sizes. 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, 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 of dimension 1.
From the point of view of the field with one element, a set can be seen as a vector space over the field with one element: this formalizes various analogies between Coxeter groups and algebraic groups. Under this correspondence, an ordering on a set corresponds to a maximal flag: an ordering is equivalent to a maximal filtration of a set. For instance, the filtration corresponds to the ordering.
