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{{Unreferenced|date=December 2009}}
In [[mathematics]], particularly in [[linear algebra]], a '''flag''' is an increasing sequence of [[Linear subspace|subspaces]] of a finite-dimensional [[vector space]] ''V''. Here "increasing" means each is a proper subspace of the next (see [[filtration (abstract algebra)|filtration]]):
:<math>\{0\} = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V.</math>
If we write the dim ''V''<sub>''i''</sub> = ''d''<sub>''i''</sub> then we have
:<math>0 = d_0 < d_1 < d_2 < \cdots < d_k = n,</math>
where ''n'' is the [[dimension (linear algebra)|dimension]] of ''V'' (assumed to be finite-dimensional). Hence, we must have ''k'' ≤ ''n''. A flag is called a '''complete flag''' if ''d''<sub>''i''</sub> = ''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''<sub>1</sub>, … ''d''<sub>''k''</sub>).

Under certain conditions the resulting sequence resembles a [[flag]] with a point connected to a line connected to a surface.

==Bases==
An ordered [[basis (linear algebra)|basis]] for ''V'' is said to be adapted to a flag if the first ''d''<sub>''i''</sub> basis vectors form a basis for ''V''<sub>''i''</sub> 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''<sub>''i''</sub> be the span of the first ''i'' basis vectors. For example, the '''{{Visible anchor|standard flag}}''' in '''R'''<sup>''n''</sup> is induced from the [[standard basis]] (''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>) where ''e''<sub>''i''</sub> denotes the vector with a 1 in the ''i''th slot and 0's elsewhere. Concretely, the standard flag is the subspaces:
:<math>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.</math>

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 <math>v_i \in V_{i-1}^\perp < V_i</math>, 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]].<ref>Harris, Joe (1991). ''Representation Theory: A First Course'', p. 95. Springer. ISBN 0387974954.</ref>

==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]] on ''V'' such that <math>T(V_i) < V_i</math> for all ''i'') is, in matrix terms, the [[algebra]] of block [[upper triangular]] matrices (with respect to an adapted basis), where the block sizes <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 [[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).

==Subspace 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]].

==Set-theoretic analogs==
{{See|Field with one element}}
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 group]]s and [[algebraic group]]s.

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) <math>\{0\} \subset \{0,1\} \subset \{0,1,2\}</math> corresponds to the ordering <math>\{0,1,2\}</math>.

==See also==
* [[Filtration (mathematics)]]
* [[Flag manifold]]
* [[Grassmannian]]

== References ==
{{Reflist}}

* {{cite book
  | last = Shafarevich
  | first = I. R. 
  | authorlink = Igor Shafarevich
  | coauthors = A. O. Remizov
  | title = Linear Algebra and Geometry
  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 2012
  | url = http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9
  | isbn = 978-3-642-30993-9}}

{{DEFAULTSORT:Flag (Linear Algebra)}}
[[Category:Linear algebra]]