Flag (linear algebra)
Encyclopedia
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
):
If we write the dim Vi = di 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 di = 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 (d1, … dk).
for V is said to be adapted to a flag if the first di basis vectors form a basis for Vi 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 Vi be the span of the first i basis vectors. For example, the in Rn is induced from the standard basis
(e1, ..., en) where ei denotes the vector with a 1 in the ith slot and 0's elsewhere. Concretely, the standard flag is the subspaces:
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
.
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 is of dimension 0, or of a vector space over
of dimension 1 (precisely the cases where only one basis exists, independently of any flag).
, 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
.
, 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)
corresponds to the ordering 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly in linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a flag is an increasing sequence of subspaces
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
of a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V. Here "increasing" means each is a proper subspace of the next (see filtration
Filtration (abstract algebra)
In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...
):
If we write the dim Vi = di 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 di = 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 (d1, … dk).
Bases
An ordered basisBasis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
for V is said to be adapted to a flag if the first di basis vectors form a basis for Vi 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 Vi be the span of the first i basis vectors. For example, the in Rn is induced from the standard basis
Standard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...
(e1, ..., en) where ei denotes the vector with a 1 in the ith slot and 0's elsewhere. Concretely, the standard flag is the subspaces:
An adapted basis is almost never unique (trivial counterexamples); see below.
A complete flag on an inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
has an essentially unique orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
: 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
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup
Maximal compact subgroup
In mathematics, a maximal compact subgroup K of a topological group G is a subgroup K that is a compact space, in the subspace topology, and maximal amongst such subgroups....
.
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
for all i) is, in matrix terms, the algebraAlgebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
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
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
(of the general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
), 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 is of dimension 0, or of a vector space over
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 analysisFunctional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the flag idea generalises to a subspace nest, namely a collection of subspaces of V that is a total order
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
for inclusion and which further is closed under arbitrary intersections and closed linear spans. See nest algebra
Nest algebra
In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by John Ringrose in the mid-1960s and have many interesting properties...
.
Set-theoretic analogs
From the point of view of the field with one elementField with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French-English pun, Fun...
, a set can be seen as a vector space over the field with one element: this formalizes various analogies between Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
s and algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
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)
corresponds to the ordering .See also
- Filtration (mathematics)
- Flag manifoldFlag manifoldIn mathematics, a generalized flag variety is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold...
- GrassmannianGrassmannianIn mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...