Flag (linear algebra)

In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a vector space "V". Here "increasing" means each is a proper subspace of the next (see filtration)::$${0} = V_0 sub V_1 sub V_2 sub cdots sub V_k = V.If we write the dim "V"_"i" = "d"_"i" then we have:$$0 = d_0 < d_1 < d_2 < cdots < d_k = n,where "n" is the dimension of "V" (assumed to be finite-dimensional). Hence, we must have "k" ≤ "n". A flag is called a complete flag if "d"_"i" = "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 ("d"₁, … "d"_"k").

Bases

An ordered basis for "V" is said to be adapted to a flag if the first "d"_"i" basis vectors form a basis for "V"_"i" 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 "V"_"i" be the span of the first "i" basis vectors. For example, the standard flag in R^"n" is induced from the standard basis ("e"₁, ..., "e"_"n") where "e"_"i" denotes the vector with a 1 in the "i"th slot and 0's elsewhere.

An adapted basis is almost never unique (trivial counterexamples); 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, like 1, -1, "i"). This is easiest to prove inductively, by noting that $$v_i in V_{i-1}^perp < V_i, 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.

tabilizer

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 on "V" such that $$T(V_i) < V_i for all "i") is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), where the block sizes $$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 is of dimension 0, or of a vector space over $$mathbf{F}_2 of dimension 1 (precisely the cases where only one basis exists, independently of any flag).

ubspace nest

In an infinite-dimensional space "V", as used in functional analysis, the flag idea generalises to a subspace nest, namely a collection of subspaces of "V" that is a total order for inclusion and which further is closed under arbitrary intersections and closed linear spans. See nest algebra.

ee also

* Flag manifold
* Grassmannian