In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration):
{ 0 } = V 0 ⊂ V 1 ⊂ V 2 ⊂ ⋯ ⊂ V k = V . {\displaystyle \{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.}
The term flag is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric. If we write that dimVi = di then we have
0 = d 0 < d 1 < d 2 < ⋯ < d k = n , {\displaystyle 0=d_{0}<d_{1}<d_{2}<\cdots <d_{k}=n,}
where n is the dimension of V (assumed to be finite). 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 (in many different ways) by inserting suitable subspaces. The signature of the flag is the sequence (d1, ..., dk).
Bases An ordered basis for V is said to be adapted to a flag V0 ⊂ V1 ⊂ ... ⊂ Vk if the first di basis 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 standard flag in Rn is induced from the standard basis (e1, ..., en) where ei denotes the vector with a 1 in the ith entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:
0 < ⟨ e 1 ⟩ < ⟨ e 1 , e 2 ⟩ < ⋯ < ⟨ e 1 , … , e n ⟩ = K n . {\displaystyle 0<\left\langle e_{1}\right\rangle <\left\langle e_{1},e_{2}\right\rangle <\cdots <\left\langle e_{1},\ldots ,e_{n}\right\rangle =K^{n}.}
An adapted basis is almost never unique (the counterexamples are trivial); 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 (scalar of unit length, e.g. 1, −1, i). Such a basis can be constructed using the Gram-Schmidt process. The uniqueness up to units follows inductively, by noting that v i {\displaystyle v_{i}} lies in the one-dimensional space V i − 1 ⊥ ∩ V i {\displaystyle V_{i-1}^{\perp }\cap V_{i}} . 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 (the linear operators T {\displaystyle T} on V such that T ( V i ) = V i {\displaystyle T(V_{i})=V_{i}} for all i) is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), where the block sizes are d i − d i − 1 {\displaystyle 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 {\displaystyle \mathbf {F} _{2}} of dimension 1 (precisely the cases where only one basis exists, independently of any flag).
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