FLAG (LINEAR ALGEBRA)

Contents

Definition

Bases

Stabilizer

Subspace nest

See also

Definition

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
:
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''₀, ''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 , 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.

Stabilizer

The stabilizer of the standard flag is the 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 $d\_i-d\_\{i-1\}$. The stabilizer of a complete flag is the upper triangular matrices (with respect to an adapted basis). In a basis adapted to a complete flag, lower triangular matrices are the stabilizer of the orthogonal flag (given by $W\_i\; =\; V\_\{n-i\}^perp$).
If we take the operators to be invertible, then we call the stabilizer of a complete flag a Borel group, and stabilizers of partial flags parabolic subgroups.
The (invertible) stabilizer acts freely transitively on adapted bases, and thus these are not unique unless the stabilizer is trivial, which happens in these degenerate cases:
the trivial flag consisting of just $\{0\}$, or the flag on a vector space over $mathbf\{F\}\_2$ with just a 1-dimensional (and optionally 0-dimensional) component.

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.

See also

★ Flag manifold

★ Grassmannian