Flag (linear algebra)

Last updated August 26, 2024

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):

Bases

An ordered basis for V is said to be adapted to a flag V₀ ⊂ V₁ ⊂ ... ⊂ 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 Rⁿ is induced from the standard basis (e₁, ..., e_n) where e_i denotes the vector with a 1 in the ith entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:

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}.

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 $v_{i}$ lies in the one-dimensional space $V_{i-1}^{\perp }\cap V_{i}$ .

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

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 $T$ 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 of dimension 0, or for a vector space over $\mathbf {F} _{2}$ 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

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) $\{0\}\subset \{0,1\}\subset \{0,1,2\}$ corresponds to the ordering $(0,1,2)$ .

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References

↑ 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.

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[1] 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.

[2] Harris, Joe (1991). Representation Theory: A First Course, p. 95. Springer. ISBN 0387974954.

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