# Flag (linear algebra)

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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):

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.^{[1]}

If we write that dim*V*_{i} = *d*_{i} then we have

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 *d*_{i} = *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 (*d*_{1}, ..., *d*_{k}).

## Bases

[edit]An ordered basis for *V* is said to be **adapted** to a flag *V*_{0} ⊂ *V*_{1} ⊂ ... ⊂ *V*_{k} 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 entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:

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 lies in the one-dimensional space .

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.^{[2]}

## Stabilizer

[edit]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 for all *i*) is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), 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 (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 of dimension 1 (precisely the cases where only one basis exists, independently of any flag).

## Subspace nest

[edit]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.

## Set-theoretic analogs

[edit]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 (flag) corresponds to the ordering .

## See also

[edit]## References

[edit]**^**Kostrikin, Alexei I. and Manin, Yuri I. (1997).*Linear Algebra and Geometry*, p. 13. Translated from the Russian by M. E. Alferieff. Gordon and Breach Science Publishers. ISBN 2-88124-683-4.**^**Harris, Joe (1991).*Representation Theory: A First Course*, p. 95. Springer. ISBN 0387974954.

- Shafarevich, I. R.; A. O. Remizov (2012).
*Linear Algebra and Geometry*. Springer. ISBN 978-3-642-30993-9.