Editing Flag (linear algebra)

{{Refimprove|date=September 2014}}
In [[mathematics]], particularly in [[linear algebra]], a '''flag''' is an increasing [[sequence]] of [[Linear subspace|subspaces]] of a [[dimension (vector space)|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>

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.<ref>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}}.</ref>

If we write that 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). Hence, we must have ''k'' ≤ ''n''. A flag is called a '''complete flag''' if ''d''<sub>''i''</sub> = ''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''<sub>1</sub>, ..., ''d''<sub>''k''</sub>).
==Bases==
An ordered [[basis (linear algebra)|basis]] for ''V'' is said to be '''adapted''' to a flag ''V''<sub>0</sub> ⊂ ''V''<sub>1</sub> ⊂ ... ⊂ ''V''<sub>''k''</sub> 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 [[linear span|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 entry and 0's elsewhere. Concretely, the standard flag is the sequence of 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 (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 [[mathematical induction|inductively]], by noting that <math>v_i</math> lies in the one-dimensional space <math>V_{i-1}^\perp \cap V_i</math>.

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 (mathematics)|group]] of [[invertible matrix|invertible]] [[upper triangular]] [[matrix (mathematics)|matrices]].

More generally, the stabilizer of a flag (the [[linear operators]] <math>T</math> on ''V'' such that <math>T(V_i) < V_i</math> for all ''i'') is, in matrix terms, the [[algebra over a field|algebra]] of block [[upper triangular]] matrices (with respect to an adapted basis), where the block sizes are <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 [[Borel subgroup#Parabolic subgroups|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 (set theory)|inclusion]] and which further is closed under arbitrary [[intersection (set theory)|intersections]] and closed linear spans. See [[nest algebra]].

==Set-theoretic analogs==
{{Further|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]]
* [[Matroid]]

== References ==
{{Reflist}}

* {{cite book
  | last = Shafarevich
  | first = I. R. 
  | authorlink = Igor Shafarevich
  |author2=A. O. Remizov
  | title = Linear Algebra and Geometry
  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 2012
  | url = https://www.springer.com/mathematics/algebra/book/978-3-642-30993-9
  | isbn = 978-3-642-30993-9}}

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