“Grassmannian”的意思、由来-开放百科全书

In mathematics, the Grassmannian {{math|Gr(k, V)}} is a space which parametrizes all {{mvar|k}}-dimensional linear subspaces of the n-dimensional vector space {{mvar|V}}. For example, the Grassmannian {{math|Gr(1, V)}} is the space of lines through the origin in {{mvar|V}}, so it is the same as the projective space of one dimension lower than {{mvar|V}}.

When {{mvar|V}} is a real or complex vector space, Grassmannians are compact smooth manifolds.^[1] In general they have the structure of a smooth algebraic variety, of dimension

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of lines in projective 3-space and parameterized them by what are now called Plücker coordinates. Grassmannians are named after Hermann Grassmann, who introduced the concept in general.

Notations vary between authors, with

being equivalent to {{math|Gr(k, V)}}, and some authors using

or {{math|Gr(k, n)}} to denote the Grassmannian of {{mvar|k}}-dimensional subspaces of an unspecified {{mvar|n}}-dimensional vector space.

Motivation

By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold {{mvar|M}} of dimension {{mvar|k}} embedded in {{math|Rⁿ}}. At each point {{mvar|x}} in {{mvar|M}}, the tangent space to {{mvar|M}} can be considered as a subspace of the tangent space of {{math|Rⁿ}}, which is just {{math|Rⁿ}}. The map assigning to {{mvar|x}} its tangent space defines a map from {{mvar|M}} to {{math|Gr(k, n)}}. (In order to do this, we have to translate the geometrical tangent space to {{mvar|M}} so that it passes through the origin rather than {{mvar|x}}, and hence defines a {{mvar|k}}-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This idea can with some effort be extended to all vector bundles over a manifold {{mvar|M}}, so that every vector bundle generates a continuous map from {{mvar|M}} to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps viewed as continuous maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. But the definition of homotopic relies on a notion of continuity, and hence a topology.

Low dimensions

For {{math|k {{=}} 1}}, the Grassmannian {{math|Gr(1, n)}} is the space of lines through the origin in n-space, so it is the same as the projective space of n−1 dimensions.

For {{math|k {{=}} 2}}, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces {{math|Gr(2, 3), Gr(1, 3), and P²}} (the projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is {{math|Gr(2, 4)}}, which may be parameterized via Plücker coordinates.

The geometric definition of the Grassmannian as a set

Let {{mvar|V}} be an

-dimensional vector space over a field {{mvar|K}}. The Grassmannian {{math|Gr(k, V)}} is the set of all {{mvar|k}}-dimensional linear subspaces of {{mvar|V}}. The Grassmannian is also denoted {{math|Gr(k, n)}} or

The Grassmannian as a differentiable manifold

To endow the Grassmannian

with the structure of a differentiable manifold, choose

a basis for

. This is equivalent to identifying it with

with the standard basis, denoted

, viewed as column vectors. Then for any

dimensional subspace

linearly independent column vectors

. The homogeneous coordinates of the element

consist of the components of the

rectangular matrix

of maximal rank whose

th column vector is

. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices

and

represent the same element

Now we define a coordinate atlas. For any

matrix

, we can apply elementary column operations to obtain its reduced column echelon form. If the first

rows of

are linearly independent, the result will have the form

The

matrix

determines

. In general, the first

rows need not be independent, but for any

whose rank is

, there exists an ordered set of integers

such that the submatrix

consisting of the

-th rows of

is nonsingular. We may apply column operations to reduce this submatrix to the identity, and the remaining entries uniquely correspond to

. Hence we have the following definition:

For each ordered set of integers

, let

be the set of

matrices

whose

submatrix

is nonsingular, where the

th row of

is the

th row of

. The coordinate function on

is then defined as the map

that sends

to the

rectangular matrix whose rows are the rows of the matrix

complementary to

. The choice of homogeneous coordinate matrix

representing the element

does not affect the values of the coordinate matrix

representing

on the coordinate neighbourhood

. Moreover, the coordinate matrices

may take arbitrary values, and they define a diffeomorphism from

onto the space of

dimensional

-valued matrices.

of any two such coordinate neighborhoods, the coordinate matrix values are related by the transition relation

The Grassmannian as a homogeneous space

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group {{math|GL(V)}} acts transitively on the {{mvar|r}}-dimensional subspaces of {{mvar|V}}. Therefore, if {{mvar|H}} is the stabilizer of any of the subspaces under this action, we have

If the underlying field is {{math|R}} or {{math|C}} and {{math|GL(V)}} is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold. It also becomes possible to use other groups to make this construction. To do this, fix an inner product on {{mvar|V}}. Over {{math|R}}, one replaces {{math|GL(V)}} by the orthogonal group {{math|O(V)}}, and by restricting to orthonormal frames, one gets the identity

Over {{math|C}}, one replaces {{math|GL(V)}} by the unitary group {{math|U(V)}}. This shows that the Grassmannian is compact. These constructions also make the Grassmannian into a metric space: For a subspace {{mvar|W}} of {{mvar|V}}, let {{math|P_W}} be the projection of {{mvar|V}} onto {{mvar|W}}. Then

where {{math|{{!!}}⋅{{!!}}}} denotes the operator norm, is a metric on {{math|Gr(r, V)}}. The exact inner product used does not matter, because a different inner product will give an equivalent norm on {{mvar|V}}, and so give an equivalent metric.

If the ground field {{mvar|k}} is arbitrary and {{math|GL(V)}} is considered as an algebraic group, then this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, {{mvar|H}} is a parabolic subgroup of {{math|GL(V)}}.

The Grassmannian as a scheme

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.^[2]

Representable functor

Let

be a quasi-coherent sheaf on a scheme {{mvar|S}}. Fix a positive integer {{mvar|r}}. Then to each {{mvar|S}}-scheme {{mvar|T}}, the Grassmannian functor associates the set of quotient modules of

This functor is representable by a separated {{mvar|S}}-scheme

. The latter is projective if

is finitely generated. When {{mvar|S}} is the spectrum of a field {{mvar|k}}, then the sheaf

is given by a vector space {{mvar|V}} and we recover the usual Grassmannian variety of the dual space of {{mvar|V}}, namely: {{math|Gr(r, V^∗)}}.

By construction, the Grassmannian scheme is compatible with base changes: for any {{mvar|S}}-scheme {{math|S′}}, we have a canonical isomorphism

In particular, for any point {{mvar|s}} of {{mvar|S}}, the canonical morphism {{math|{s} {{=}} Spec(k(s)) → S}}, induces an isomorphism from the fiber

to the usual Grassmannian

over the residue field {{math|k(s)}}.

Universal family

Since the Grassmannian scheme represents a functor, it comes with a universal object,

, which is an object of

and therefore a quotient module

, locally free of rank {{mvar|r}} over

. The quotient homomorphism induces a closed immersion from the projective bundle

Conversely, any such closed immersion comes from a surjective homomorphism of {{math|O_T}}-modules from

to a locally free module of rank {{mvar|r}}.^[3] Therefore, the elements of

are exactly the projective subbundles of rank {{mvar|r}} in

Under this identification, when {{math|T {{=}} S}} is the spectrum of a field {{mvar|k}} and

is given by a vector space {{mvar|V}}, the set of rational points

correspond to the projective linear subspaces of dimension {{math|r − 1}} in {{math|P(V)}}, and the image of

The Plücker embedding

The Plücker embedding is a natural embedding of the Grassmannian

into the projectivization of the exterior algebra

Suppose that {{mvar|W}} is a {{mvar|k}}-dimensional subspace of the

dimensional vector space {{mvar|V}}. To define

, choose a basis {{math|{w₁, ..., w_k},}} of {{mvar|W}}, and let

be the wedge product of these basis elements:

A different basis for {{mvar|W}} will give a different wedge product, but the two products will differ only by a non-zero scalar (the determinant of the change of basis matrix). Since the right-hand side takes values in a projective space,

is well-defined. To see that

is an embedding, notice that it is possible to recover

from

as the span of the set of all vectors

such that

Plücker coordinates and the Plücker relations

The Plücker embedding of the Grassmannian satisfies some very simple quadratic relations called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of {{math|P(∧^kV)}} and give another method of constructing the Grassmannian. To state the Plücker relations, fix a basis {{math|{e₁, ..., e_n}}} of {{mvar|V}}, and let {{mvar|W}} be a {{mvar|k}}-dimensional subspace of {{mvar|V}} with basis {{math|{w₁, ..., w_k}}}. Let {{math|(w_i1, ..., w_in)}} be the coordinates of {{math|w_i}} with respect to the basis {{math|{e₁, ..., e_n}}} of {{mvar|V}}, let

and let {{math|{W₁, ..., W_n}}} be the columns of

. For any ordered sequence

positive integers, let

be the determinant of the

matrix with columns

. The set

is called the Plücker coordinates of the element

of the Grassmannian (with respect to the basis {{math|{e₁, ..., e_n}}} of {{mvar|V}}). They are the linear coordinates of the image

under the Plücker map, relative to the basis of the exterior power

induced by the basis {{math|{e₁, ..., e_n}}} of {{mvar|V}}.

For any two ordered sequences

and

positive integers, resp., the following homogeneous equations are valid and determine the image of {{mvar|W}} under the Plücker embedding:

When {{math|dim(V) {{=}} 4}}, and {{math|k {{=}} 2}}, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of {{math|P(∧^kV)}} by {{math|W₁₂, W₁₃, W₁₄, W₂₃, W₂₄, W₃₄}}, the image of {{math|Gr(2, V)}} under the Plücker map is defined by the single equation

In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.^[4]

The Grassmannian as a real affine algebraic variety

Let {{math|Gr(r, Rⁿ)}} denote the Grassmannian of {{mvar|r}}-dimensional subspaces of {{math|Rⁿ}}. Let {{math|M(n, R)}} denote the space of real {{math|n × n}} matrices. Consider the set of matrices {{math|A(r, n) ⊂ M(n, R)}} defined by {{math|X ∈ A(r, n)}} if and only if the three conditions are satisfied:

Duality

Every {{mvar|r}}-dimensional subspace {{mvar|W}} of {{mvar|V}} determines an {{math|(n – r)}}-dimensional quotient space {{math|V/W}} of {{mvar|V}}. This gives the natural short exact sequence:

Taking the dual to each of these three spaces and linear transformations yields an inclusion of {{math|(V/W)^∗}} in {{math|V^∗}} with quotient {{math|W^∗}}:

Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between {{mvar|r}}-dimensional subspaces of {{mvar|V}} and {{math|(n – r)}}-dimensional subspaces of {{math|V^∗}}. In terms of the Grassmannian, this is a canonical isomorphism

Choosing an isomorphism of {{mvar|V}} with {{math|V^∗}} therefore determines a (non-canonical) isomorphism of {{math|Gr(r, V)}} and {{math|Gr(n − r, V)}}. An isomorphism of {{mvar|V}} with {{math|V^∗}} is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an {{mvar|r}}-dimensional subspace into its {{math|(n – r)}}-dimensional orthogonal complement.

Schubert cells

The detailed study of the Grassmannians uses a decomposition into subsets called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for {{math|Gr(r, n)}} are defined in terms of an auxiliary flag: take subspaces {{math|V₁, V₂, ..., V_r}}, with {{math|V_i ⊂ V_{i + 1}}}. Then we consider the corresponding subset of {{math|Gr(r, n)}}, consisting of the {{mvar|W}} having intersection with {{math|V_i}} of dimension at least {{mvar|i}}, for {{math|i {{=}} 1, ..., r}}. The manipulation of Schubert cells is Schubert calculus.

Here is an example of the technique. Consider the problem of determining the Euler characteristic of the Grassmannian of {{mvar|r}}-dimensional subspaces of {{math|Rⁿ}}. Fix a {{math|1}}-dimensional subspace {{math|R ⊂ Rⁿ}} and consider the partition of {{math|Gr(r, n)}} into those {{mvar|r}}-dimensional subspaces of {{math|Rⁿ}} that contain {{math|R}} and those that do not. The former is {{math|Gr(r − 1, n − 1)}} and the latter is a {{mvar|r}}-dimensional vector bundle over {{math|Gr(r, n − 1)}}. This gives recursive formulas:

If one solves this recurrence relation, one gets the formula: {{math|χ_{r, n} {{=}} 0}} if and only if {{mvar|n}} is even and {{mvar|r}} is odd. Otherwise:

Cohomology ring of the complex Grassmannian

Every point in the complex Grassmannian manifold {{math|Gr(r, n)}} defines an {{mvar|r}}-plane in {{mvar|n}}-space. Fibering these planes over the Grassmannian one arrives at the vector bundle {{mvar|E}} which generalizes the tautological bundle of a projective space. Similarly the {{math|(n − r)}}-dimensional orthogonal complements of these planes yield an orthogonal vector bundle {{mvar|F}}. The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of {{mvar|E}}. In particular, all of the integral cohomology is at even degree as in the case of a projective space.

These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of {{mvar|E}} and {{mvar|F}}. Then the relations merely state that the direct sum of the bundles {{mvar|E}} and {{mvar|F}} is trivial. Functoriality of the total Chern classes allows one to write this relation as

The quantum cohomology ring was calculated by Edward Witten in [https://arxiv.org/abs/hep-th/9312104 The Verlinde Algebra And The Cohomology Of The Grassmannian]. The generators are identical to those of the classical cohomology ring, but the top relation is changed to

reflecting the existence in the corresponding quantum field theory of an instanton with {{math|2n}} fermionic zero-modes which violates the degree of the cohomology corresponding to a state by {{math|2n}} units.

Associated measure

When {{mvar|V}} is {{mvar|n}}-dimensional Euclidean space, one may define a uniform measure on {{math|Gr(r, n)}} in the following way. Let {{math|θ_n}} be the unit Haar measure on the orthogonal group {{math|O(n)}} and fix {{mvar|V}} in {{math|Gr(r, n)}}. Then for a set {{math|A ⊆ Gr(r, n)}}, define

This measure is invariant under actions from the group {{math|O(n)}}, that is, {{math|γ_{r, n}(gA) {{=}} γ_{r, n}(A)}} for all {{mvar|g}} in {{math|O(n)}}. Since {{math|θ_n(O(n)) {{=}} 1}}, we have {{math|γ_{r, n}(Gr(r, n)) {{=}} 1}}. Moreover, {{math|γ_{r, n}}} is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.

Oriented Grassmannian

This is the manifold consisting of all oriented {{mvar|r}}-dimensional subspaces of {{math|Rⁿ}}. It is a double cover of {{math|Gr(r, n)}} and is denoted by:

Applications

Grassmann manifolds have found application in computer vision tasks of video-based face recognition and shape recognition.^[5] They are also used in the data-visualization technique known as the grand tour.

Grassmannians allow the scattering amplitudes of subatomic particles to be calculated via a positive Grassmannian construct called the amplituhedron.^[6]