Stiefel manifold

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From Wikipedia, the free encyclopedia

In mathematics, the Stiefel manifold

𝑉𝑘(𝑅𝑛)

is the set of all orthonormal k-frames in

𝑅𝑛.

That is, it is the set of ordered orthonormal k-tuples of vectors in

𝑅𝑛.

It is named after Swiss mathematician Eduard Stiefel. Likewise one can

define the complex Stiefel manifold

𝑉𝑘(𝐶𝑛)

of orthonormal k-frames in

𝐶𝑛

and the quaternionic Stiefel manifold

𝑉𝑘(𝐻𝑛)
 of orthonormal k-frames in

𝐻𝑛

. More generally, the construction applies to any real, complex, or quaternionic

inner product space.

In some contexts, a non-compact Stiefel manifold is defined as the set of all

linearly independent k-frames in

𝑅𝑛,𝐶𝑛,

or

𝐻𝑛;

this is homotopy equivalent, as the compact Stiefel manifold is a deformation

retract of the non-compact one, by Gram–Schmidt. Statements about the non-
compact form correspond to those for the compact form, replacing the
orthogonal group (or unitary or symplectic group) with the general linear group.

Topology[edit]

Let

stand for

𝑅,𝐶,

or

𝐻.

The Stiefel manifold

𝑉𝑘(𝐹𝑛)

can be thought of as a set of n × k matrices by writing a k-frame as a

matrix of k column vectors in

𝐹𝑛.

The orthonormality condition is expressed by A*A =

𝐼𝑘

where A* denotes the conjugate transpose of A and

𝐼𝑘

denotes the k × k identity matrix. We then have

𝑉𝑘(𝐹𝑛)={𝐴∈𝐹𝑛×𝑘:𝐴∗𝐴=𝐼𝑘}.

The topology on

𝑉𝑘(𝐹𝑛)

is the subspace topology inherited from

𝐹𝑛×𝑘.

With this topology

𝑉𝑘(𝐹𝑛)

is a compact manifold whose dimension is given by

dim⁡𝑉𝑘(𝑅𝑛)=𝑛𝑘−12𝑘(𝑘+1)dim⁡𝑉𝑘(𝐶𝑛)=2𝑛𝑘−𝑘2dim⁡𝑉𝑘(𝐻𝑛)=4𝑛𝑘−𝑘(2𝑘−1)
As a homogeneous space[edit]

Each of the Stiefel manifolds

𝑉𝑘(𝐹𝑛)

can be viewed as a homogeneous space for the action of a classical

group in a natural manner.

Every orthogonal transformation of a k-frame in

𝑅𝑛

results in another k-frame, and any two k-frames are related by some
orthogonal transformation. In other words, the orthogonal group O(n) acts
transitively on

𝑉𝑘(𝑅𝑛).

The stabilizer subgroup of a given frame is the subgroup isomorphic to

O(n−k) which acts nontrivially on the orthogonal complement of the space
spanned by that frame.

Likewise the unitary group U(n) acts transitively on

𝑉𝑘(𝐶𝑛)

with stabilizer subgroup U(n−k) and the symplectic group Sp(n) acts
transitively on

𝑉𝑘(𝐻𝑛)
 with stabilizer subgroup Sp(n−k).

In each case

𝑉𝑘(𝐹𝑛)

can be viewed as a homogeneous space:

𝑉𝑘(𝑅𝑛)≅O(𝑛)/O(𝑛−𝑘)𝑉𝑘(𝐶𝑛)≅U(𝑛)/U(𝑛−𝑘)𝑉𝑘(𝐻𝑛)≅Sp(𝑛)/Sp(𝑛−𝑘)

When k = n, the corresponding action is free so that the Stiefel manifold

𝑉𝑛(𝐹𝑛)

is a principal homogeneous space for the corresponding classical group.

When k is strictly less than n then the special orthogonal group SO(n) also acts
transitively on

𝑉𝑘(𝑅𝑛)

with stabilizer subgroup isomorphic to SO(n−k) so that

𝑉𝑘(𝑅𝑛)≅SO(𝑛)/SO(𝑛−𝑘)for 𝑘<𝑛.

The same holds for the action of the special unitary group on

𝑉𝑘(𝐶𝑛)

𝑉𝑘(𝐶𝑛)≅SU(𝑛)/SU(𝑛−𝑘)for 𝑘<𝑛.
Thus for k = n − 1, the Stiefel manifold is a principal homogeneous space for the
corresponding special classical group.

Uniform measure[edit]

The Stiefel manifold can be equipped with a uniform measure, i.e. a Borel
measure that is invariant under the action of the groups noted above. For
example,

𝑉1(𝑅2)

which is isomorphic to the unit circle in the Euclidean plane, has as its
uniform measure the obvious uniform measure (arc length) on the circle. It is
straightforward to sample this measure on

𝑉𝑘(𝐹𝑛)

using Gaussian random matrices: if

𝐴∈𝐹𝑛×𝑘

is a random matrix with independent entries identically distributed

according to the standard normal distribution on

and A = QR is the QR factorization of A, then the matrices,

𝑄∈𝐹𝑛×𝑘,𝑅∈𝐹𝑘×𝑘

are independent random variables and Q is distributed

according to the uniform measure on

𝑉𝑘(𝐹𝑛).
 This result is a consequence of the Bartlett decomposition theorem.[1]

Special cases[edit]

A 1-frame in

𝐹𝑛

is nothing but a unit vector, so the Stiefel manifold

𝑉1(𝐹𝑛)

is just the unit sphere in

𝐹𝑛.

Therefore:

𝑉1(𝑅𝑛)=𝑆𝑛−1𝑉1(𝐶𝑛)=𝑆2𝑛−1𝑉1(𝐻𝑛)=𝑆4𝑛−1

Given a 2-frame in

𝑅𝑛,

let the first vector define a point in Sn−1 and the second a unit tangent vector
to the sphere at that point. In this way, the Stiefel manifold

𝑉2(𝑅𝑛)

may be identified with the unit tangent bundle to Sn−1.

When k = n or n−1 we saw in the previous section that

𝑉𝑘(𝐹𝑛)

is a principal homogeneous space, and therefore diffeomorphic to the

corresponding classical group:

𝑉𝑛−1(𝑅𝑛)≅SO(𝑛)𝑉𝑛−1(𝐶𝑛)≅SU(𝑛)

𝑉𝑛(𝑅𝑛)≅O(𝑛)𝑉𝑛(𝐶𝑛)≅U(𝑛)𝑉𝑛(𝐻𝑛)≅Sp(𝑛)

Functoriality[edit]

Given an orthogonal inclusion between vector spaces

𝑋↪𝑌,

the image of a set of k orthonormal vectors is orthonormal, so there is

an induced closed inclusion of Stiefel manifolds,

𝑉𝑘(𝑋)↪𝑉𝑘(𝑌),

and this is functorial. More subtly, given an n-dimensional

vector space X, the dual basis construction gives a bijection between bases for X
and bases for the dual space

𝑋∗,

which is continuous, and thus yields a homeomorphism of top Stiefel

manifolds
𝑉𝑛(𝑋)→∼𝑉𝑛(𝑋∗).

This is also functorial for isomorphisms of vector spaces.

As a principal bundle[edit]

There is a natural projection

𝑝:𝑉𝑘(𝐹𝑛)→𝐺𝑘(𝐹𝑛)

from the Stiefel manifold

𝑉𝑘(𝐹𝑛)

to the Grassmannian of k-planes in

𝐹𝑛

which sends a k-frame to the subspace spanned by that frame. The fiber over
a given point P in

𝐺𝑘(𝐹𝑛)

is the set of all orthonormal k-frames contained in the space P.

This projection has the structure of a principal G-bundle where G is the

associated classical group of degree k. Take the real case for concreteness.
There is a natural right action of O(k) on

𝑉𝑘(𝑅𝑛)

which rotates a k-frame in the space it spans. This action is free but not
transitive. The orbits of this action are precisely the orthonormal k-frames
spanning a given k-dimensional subspace; that is, they are the fibers of the map
p. Similar arguments hold in the complex and quaternionic cases.
We then have a sequence of principal bundles:

O(𝑘)→𝑉𝑘(𝑅𝑛)→𝐺𝑘(𝑅𝑛)U(𝑘)→𝑉𝑘(𝐶𝑛)→𝐺𝑘(𝐶𝑛)Sp(𝑘)→𝑉𝑘(𝐻𝑛)→𝐺𝑘(𝐻𝑛)

The vector bundles associated to these principal bundles via the natural action of
G on

𝐹𝑘

are just the tautological bundles over the Grassmannians. In other words, the
Stiefel manifold

𝑉𝑘(𝐹𝑛)

is the orthogonal, unitary, or symplectic frame bundle associated to the

tautological bundle on a Grassmannian.

When one passes to the

𝑛→∞

limit, these bundles become the universal bundles for the classical
groups.

Homotopy[edit]

The Stiefel manifolds fit into a family of fibrations:

𝑉𝑘−1(𝑅𝑛−1)→𝑉𝑘(𝑅𝑛)→𝑆𝑛−1,

thus the first non-trivial homotopy group of the space

𝑉𝑘(𝑅𝑛)

is in dimension n − k. Moreover,

𝜋𝑛−𝑘𝑉𝑘(𝑅𝑛)≃{𝑍𝑛−𝑘 even or 𝑘=1𝑍2𝑛−𝑘 odd and 𝑘>1

This result is used in the obstruction-theoretic definition of Stiefel–Whitney

classes.

See also[edit]

● Flag manifold
● Matrix Langevin distribution[2][3]

References[edit]

● ^ Muirhead, Robb J. (1982). Aspects of Multivariate Statistical Theory. John Wiley &
Sons, Inc., New York. pp. xix+673. ISBN 0-471-09442-0.
● ^ Chikuse, Yasuko (1 May 2003). "Concentrated matrix Langevin distributions". Journal
of Multivariate Analysis. 85 (2): 375–394. doi:10.1016/S0047-259X(02)00065-9. ISSN 0047-
259X.
● ^ Pal, Subhadip; Sengupta, Subhajit; Mitra, Riten; Banerjee, Arunava (September 2020).
"Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the
Stiefel Manifold". Bayesian Analysis. 15 (3): 871–908. doi:10.1214/19-BA1176. ISSN 1936-
0975.
● Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press.
ISBN 0-521-79540-0.
● Husemoller, Dale (1994). Fibre Bundles ((3rd ed.) ed.). New York:
Springer-Verlag. ISBN 0-387-94087-1.
● James, Ioan Mackenzie (1976). The topology of Stiefel manifolds. CUP
Archive. ISBN 978-0-521-21334-9.
● "Stiefel manifold", Encyclopedia of Mathematics, EMS Press, 2001
[1994]

Categories:

Differential geometry
Homogeneous spaces
 Fiber bundles
Manifolds
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Grassmannian
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From Wikipedia, the free encyclopedia

For other uses, see Grassmannian (disambiguation).

In mathematics, the Grassmannian

𝐺𝑟𝑘(𝑉)

(named in honour of Hermann Grassmann) is a differentiable manifold

that parameterizes the set of all

-dimensional linear subspaces of an

-dimensional vector space

𝑉
 over a field

. For example, the Grassmannian

𝐺𝑟1(𝑉)

is the space of lines through the origin in

, so it is the same as the projective space

𝑃(𝑉)

of one dimension lower than

.[1][2] When

is a real or complex vector space, Grassmannians are compact smooth

manifolds, of dimension

𝑘(𝑛−𝑘)

.[3] In general they have the structure of a nonsingular projective

algebraic variety.

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who

studied the set of projective lines in real projective 3-space, which is equivalent
to

𝐺𝑟2(𝑅4)
 , parameterizing them by what are now called Plücker coordinates. (See
§ Plücker coordinates and Plücker relations below.) Hermann Grassmann later
introduced the concept in general.

Notations for Grassmannians vary between authors, and include

𝐺𝑟𝑘(𝑉)

𝐺𝑟(𝑘,𝑉)

𝐺𝑟𝑘(𝑛)

𝐺𝑟(𝑘,𝑛)

to denote the Grassmannian of

-dimensional subspaces of an

-dimensional vector space

Motivation[edit]
By giving a collection of subspaces of a vector space a topological structure, it is
possible to talk about a continuous choice of subspaces or open and closed
collections of subspaces. Giving them the further 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 a Euclidean space. Suppose we have a manifold

of dimension

embedded in

𝑅𝑛

. At each point

𝑥∈𝑀

, the tangent space to

can be considered as a subspace of the tangent space of

𝑅𝑛

, which is also just

𝑅𝑛

. The map assigning to

its tangent space defines a map from M to

𝐺𝑟𝑘(𝑅𝑛)

. (In order to do this, we have to translate the tangent space at each

𝑥∈𝑀

so that it passes through the origin rather than

, and hence defines a

-dimensional vector subspace. This idea is very similar to the Gauss map for
surfaces in a 3-dimensional space.)

This can with some effort be extended to all vector bundles over a manifold

, so that every vector bundle generates a continuous map from

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. In
particular we find that vector bundles inducing homotopic maps to the
Grassmannian are isomorphic. Here the definition of homotopy relies on a notion
of continuity, and hence a topology.

Low dimensions[edit]

For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-
space, so it is the same as the projective space

𝑃𝑛−1
 of n − 1 dimensions.

For 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
2
that plane (and vice versa); hence the spaces Gr(2, 3), Gr(1, 3), and P (the
projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is Gr(2, 4).

The Grassmannian as a differentiable manifold[edit]

To endow

𝐺𝑟𝑘(𝑉)

with the structure of a differentiable manifold, choose a basis for

. This is equivalent to identifying

with

𝐾𝑛

, with the standard basis denoted

(𝑒1,…,𝑒𝑛)

, viewed as column vectors. Then for any

𝑘
 -dimensional subspace

𝑤⊂𝑉

, viewed as an element of

𝐺𝑟𝑘(𝑉)

, we may choose a basis consisting of

linearly independent column vectors

(𝑊1,…,𝑊𝑘)

. The homogeneous coordinates of the element

𝑤∈𝐺𝑟𝑘(𝑉)

consist of the elements of the

𝑛×𝑘

maximal rank rectangular matrix

whose

-th column vector is

𝑊𝑖

,
𝑖=1,…,𝑘

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

rectangular matrices

and

𝑊~

represent the same element

𝑤∈𝐺𝑟𝑘(𝑉)

if and only if

𝑊~=𝑊𝑔

for some element

𝑔∈𝐺𝐿(𝑘,𝐾)

of the general linear group of invertible

𝑘×𝑘

matrices with entries in

. This defines an equivalence relation between

𝑛×𝑘

matrices
𝑊

of rank

, for which the equivalence classes are denoted

[𝑊]

We now define a coordinate atlas. For any

𝑛×𝑘

homogeneous coordinate matrix

, we can apply elementary column operations (which amounts to multiplying

by a sequence of elements

𝑔∈𝐺𝐿(𝑘,𝐾)

) to obtain its reduced column echelon form. If the first

rows of

are linearly independent, the result will have the form

 [11⋱1𝑎1,1⋯⋯𝑎1,𝑘⋮⋮𝑎𝑛−𝑘,1⋯⋯𝑎𝑛−𝑘,𝑘]

and the

(𝑛−𝑘)×𝑘

affine coordinate matrix

with entries

(𝑎𝑖𝑗)

determines

. In general, the first

rows need not be independent, but since

has maximal rank

𝑘

, there exists an ordered set of integers

1≤𝑖1<⋯<𝑖𝑘≤𝑛

such that the

𝑘×𝑘

submatrix

𝑊𝑖1,…,𝑖𝑘

whose rows are the

(𝑖1,…,𝑖𝑘)

-th rows of

is nonsingular. We may apply column operations to reduce this submatrix to

the identity matrix, and the remaining entries uniquely determine

. Hence we have the following definition:

For each ordered set of integers

1≤𝑖1<⋯<𝑖𝑘≤𝑛

, let

𝑈𝑖1,…,𝑖𝑘

be the set of elements

𝑤∈𝐺𝑟𝑘(𝑉)

for which, for any choice of homogeneous coordinate matrix

, the

𝑘×𝑘

submatrix

𝑊𝑖1,…,𝑖𝑘

whose

-th row is the

𝑖𝑗

-th row of

is nonsingular. The affine coordinate functions on

𝑈𝑖1,…,𝑖𝑘

are then defined as the entries of the

(𝑛−𝑘)×𝑘

matrix

𝐴𝑖1,…,𝑖𝑘
 whose rows are those of the matrix

𝑊𝑊𝑖1,…,𝑖𝑘−1

complementary to

(𝑖1,…,𝑖𝑘)

, written in the same order. The choice of homogeneous

𝑛×𝑘

coordinate matrix

in

[𝑊]

representing the element

𝑤∈𝐺𝑟𝑘(𝑉)

does not affect the values of the affine coordinate matrix

𝐴𝑖1,…,𝑖𝑘

representing w on the coordinate neighbourhood

𝑈𝑖1,…,𝑖𝑘

. Moreover, the coordinate matrices

𝐴𝑖1,…,𝑖𝑘
 may take arbitrary values, and they define a diffeomorphism from

𝑈𝑖1,…,𝑖𝑘

to the space of

-valued

(𝑛−𝑘)×𝑘

matrices. Denote by

𝐴^𝑖1,…,𝑖𝑘:=𝑊(𝑊𝑖1,…,𝑖𝑘)−1

the homogeneous coordinate matrix having the identity matrix as the

𝑘×𝑘

submatrix with rows

(𝑖1,…,𝑖𝑘)

and the affine coordinate matrix

𝐴𝑖1,…,𝑖𝑘

in the consecutive complementary rows. On the overlap

𝑈𝑖1,…,𝑖𝑘∩𝑈𝑗1,…,𝑗𝑘

between any two such coordinate neighborhoods, the affine

coordinate matrix values
𝐴𝑖1,…,𝑖𝑘

and

𝐴𝑗1,…,𝑗𝑘

are related by the transition relations

𝐴^𝑖1,…,𝑖𝑘𝑊𝑖1,…,𝑖𝑘=𝐴^𝑗1,…,𝑗𝑘𝑊𝑗1,…,𝑗𝑘,

where both

𝑊𝑖1,…,𝑖𝑘

and

𝑊𝑗1,…,𝑗𝑘

are invertible. This may equivalently be written as

𝐴^𝑗1,…,𝑗𝑘=𝐴^𝑖1,…,𝑖𝑘(𝐴^𝑗1,…,𝑗𝑘𝑖1,…,𝑖𝑘)−1,

where

𝐴^𝑗1,…,𝑗𝑘𝑖1,…,𝑖𝑘

is the invertible

𝑘×𝑘

matrix whose
𝑙

th row is the

𝑗𝑙

th row of

𝐴^𝑖1,…,𝑖𝑘

. The transition functions are therefore rational in the matrix elements of

𝐴𝑖1,…,𝑖𝑘

, and

{𝑈𝑖1,…,𝑖𝑘,𝐴𝑖1,…,𝑖𝑘}

gives an atlas for

𝐺𝑟𝑘(𝑉)

as a differentiable manifold and also as an algebraic variety.

The Grassmannian as a set of orthogonal projections[edit]

An alternative way to define a real or complex Grassmannian as a manifold is to

view it as a set of orthogonal projection operators (Milnor & Stasheff (1974)
problem 5-C). For this, choose a positive definite real or Hermitian inner product

⟨⋅,⋅⟩

on

𝑉
 , depending on whether

is real or complex. A

-dimensional subspace

determines a unique orthogonal projection operator

𝑃𝑤:𝑉→𝑉

whose image is

𝑤⊂𝑉

by splitting

into the orthogonal direct sum

𝑉=𝑤⊕𝑤⊥

of

and its orthogonal complement

𝑤⊥
 and defining

𝑃𝑤(𝑣)={𝑣 if 𝑣∈𝑤0 if 𝑣∈𝑤⊥.

Conversely, every projection operator

of rank

defines a subspace

𝑤𝑃:=Im(𝑃)

as its image. Since the rank of an orthogonal projection operator

equals its trace, we can identify the Grassmann manifold

𝐺𝑟(𝑘,𝑉)

with the set of rank

orthogonal projection operators

𝐺𝑟(𝑘,𝑉)∼{𝑃∈End(𝑉)∣𝑃=𝑃2=𝑃†,tr(𝑃)=𝑘}.
In particular, taking

𝑉=𝑅𝑛

or

𝑉=𝐶𝑛

this gives completely explicit equations for embedding the

Grassmannians

𝐺𝑟(𝑘,𝑅𝑁)

𝐺𝑟(𝑘,𝐶𝑁)

in the space of real or complex

𝑛×𝑛

matrices

𝑅𝑛×𝑛

𝐶𝑛×𝑛

, respectively.

Since this defines the Grassmannian as a closed subset of the sphere

{𝑋∈End(𝑉)∣tr(𝑋𝑋†)=𝑘}

this is one way to see that the Grassmannian is a

compact Hausdorff space. This construction also turns the Grassmannian
𝐺𝑟(𝑘,𝑉)

into a metric space with metric

𝑑(𝑤,𝑤′):=‖𝑃𝑤−𝑃𝑤′‖,

for any pair

𝑤,𝑤′⊂𝑉

of

-dimensional subspaces, where ‖⋅‖ denotes the operator norm. The exact inner
product used does not matter, because a different inner product will give an
equivalent norm on

, and hence an equivalent metric.

For the case of real or complex Grassmannians, the following is an equivalent

way to express the above construction in terms of matrices.

Grassmannians

𝐺𝑟(𝑘,𝑅𝑛)

𝐺𝑟(𝑘,𝐶𝑛)
 as affine algebraic varieties[edit]

Let

𝑀(𝑛,𝑅)

denote the space of real

𝑛×𝑛

matrices and the subset

𝑃(𝑘,𝑛,𝑅)⊂𝑀(𝑛,𝑅)

of matrices

𝑃∈𝑀(𝑛,𝑅)

that satisfy the three conditions:

● 𝑃
● is a projection operator:
● 𝑃2=𝑃
● .
● 𝑃
● is symmetric:
● 𝑃𝑇=𝑃
● .
● 𝑃
● has trace
● tr⁡(𝑃)=𝑘
● .

There is a bijective correspondence between

𝑃(𝑘,𝑛,𝑅)
 and the Grassmannian

𝐺𝑟(𝑘,𝑅𝑛)

of

-dimensional subspaces of

𝑅𝑛

given by sending

𝑃∈𝑃(𝑘,𝑛,𝑅)

to the

-dimensional subspace of

𝑅𝑛

spanned by its columns and, conversely, sending any element

𝑤∈𝐺𝑟(𝑘,𝑅𝑛)

to the projection matrix

𝑃𝑤:=∑𝑖=1𝑘𝑤𝑖𝑤𝑖𝑇,

where
(𝑤1,⋯,𝑤𝑘)

is any orthonormal basis for

𝑤⊂𝑅𝑛

, viewed as real

component column vectors.

An analogous construction applies to the complex Grassmannian

𝐺𝑟(𝑘,𝐶𝑛)

, identifying it bijectively with the subset

𝑃(𝑘,𝑛,𝐶)⊂𝑀(𝑛,𝐶)

of complex

𝑛×𝑛

matrices

𝑃∈𝑀(𝑛,𝐶)

satisfying

● 𝑃
● is a projection operator:
● 𝑃2=𝑃
● .
● 𝑃
● is self-adjoint (Hermitian):
● 𝑃†=𝑃
● .
● 𝑃
 ● has trace
● tr(P)=k
● ,

where the self-adjointness is with respect to the Hermitian inner product

⟨⋅,⋅⟩

in which the standard basis vectors

(𝑒1,⋯,𝑒𝑛)

are orthonomal. The formula for the orthogonal projection matrix

𝑃𝑤

onto the complex

-dimensional subspace

𝑤⊂𝐶𝑛

spanned by the orthonormal (unitary) basis vectors

(𝑤1,⋯,𝑤𝑘)

is

𝑃𝑤:=∑𝑖=1𝑘𝑤𝑖𝑤𝑖†.

The Grassmannian as a homogeneous space[edit]

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

GL(𝑉)

acts transitively on the

-dimensional subspaces of

. Therefore, if we choose a subspace

𝑤0⊂𝑉

of dimension

, any element

𝑤∈𝐺𝑟(𝑘,𝑉)

can be expressed as

𝑤=𝑔(𝑤0)

for some group element

𝑔∈GL(𝑉)

, where

𝑔
 is determined only up to right multiplication by elements

{ℎ∈𝐻}

of the stabilizer of

𝑤0

𝐻:=stab(𝑤0):={ℎ∈GL(𝑉)|ℎ(𝑤0)=𝑤0}⊂GL(𝑉)

under the

GL(𝑉)

-action.

We may therefore identify

𝐺𝑟(𝑘,𝑉)

with the quotient space

𝐺𝑟(𝑘,𝑉)=GL(𝑉)/𝐻

of left cosets of

If the underlying field is

𝑅

or

and

GL(𝑉)

is considered as a Lie group, this construction makes the Grassmannian

a smooth manifold under the quotient structure. More generally, over a ground
field

, the group

GL(𝑉)

is an algebraic group, and 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,

is a parabolic subgroup of

GL(𝑉)

Over

or
𝐶

it also becomes possible to use smaller groups in this construction. To do this

over

, fix a Euclidean inner product

on

. The real orthogonal group

𝑂(𝑉,𝑞)

acts transitively on the set of

-dimensional subspaces

𝐺𝑟(𝑘,𝑉)

and the stabiliser of a

-space

𝑤0⊂𝑉

is

𝑂(𝑤0,𝑞|𝑤0)×𝑂(𝑤0⊥,𝑞|𝑤0⊥)
 ,

where

𝑤0⊥

is the orthogonal complement of

𝑤0

in

. This gives an identification as the homogeneous space

𝐺𝑟(𝑘,𝑉)=𝑂(𝑉,𝑞)/(𝑂(𝑤,𝑞|𝑤)×𝑂(𝑤⊥,𝑞|𝑤⊥))

If we take

𝑉=𝑅𝑛

and

𝑤0=𝑅𝑘⊂𝑅𝑛

(the first

components) we get the isomorphism

𝐺𝑟(𝑘,𝑅𝑛)=𝑂(𝑛)/(𝑂(𝑘)×𝑂(𝑛−𝑘)).
Over C, if we choose an Hermitian inner product

, the unitary group

𝑈(𝑉,ℎ)

acts transitively, and we find analogously

𝐺𝑟(𝑘,𝑉)=𝑈(𝑉,ℎ)/(𝑈(𝑤0,ℎ|𝑤0)×𝑈(𝑤0⊥|,ℎ𝑤0⊥)),

or, for

𝑉=𝐶𝑛

and

𝑤0=𝐶𝑘⊂𝐶𝑛

𝐺𝑟(𝑘,𝐶𝑛)=𝑈(𝑛)/(𝑈(𝑘)×𝑈(𝑛−𝑘)).

In particular, this shows that the Grassmannian is compact, and of (real or

complex) dimension k(n − k).

The Grassmannian as a scheme[edit]

In the realm of algebraic geometry, the Grassmannian can be constructed as a

scheme by expressing it as a representable functor.[4]
Representable functor[edit]

Let

be a quasi-coherent sheaf on a scheme

. Fix a positive integer

. Then to each

-scheme

, the Grassmannian functor associates the set of quotient modules of

𝐸𝑇:=𝐸⊗𝑂𝑆𝑂𝑇

locally free of rank

on

. We denote this set by

𝐺𝑟(𝑘,𝐸𝑇)

This functor is representable by a separated

-scheme

𝐺𝑟(𝑘,𝐸)

. The latter is projective if

is finitely generated. When

is the spectrum of a field

, then the sheaf

is given by a vector space

and we recover the usual Grassmannian variety of the dual space of

, namely:
𝐺𝑟(𝑘,𝑉)

. By construction, the Grassmannian scheme is compatible with base

changes: for any

-scheme

𝑆′

, we have a canonical isomorphism

𝐺𝑟(𝑘,𝐸)×𝑆𝑆′≃𝐺𝑟(𝑘,𝐸𝑆′)

In particular, for any point

of

, the canonical morphism

{𝑠}=Spec𝐾(𝑠)→𝑆

induces an isomorphism from the fiber

𝐺𝑟(𝑘,𝐸)𝑠

to the usual Grassmannian

𝐺𝑟(𝑘,𝐸⊗𝑂𝑆𝐾(𝑠))
 over the residue field

𝐾(𝑠)

Universal family[edit]

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

object,

, which is an object of

𝐺𝑟(𝑘,𝐸𝐺𝑟(𝑘,𝐸)),

and therefore a quotient module

of

𝐸𝐺𝑟(𝑘,𝐸)

, locally free of rank

over

𝐺𝑟(𝑘,𝐸)

. The quotient homomorphism induces a closed immersion from the

projective bundle:

𝑃(𝐺)→𝑃(𝐸𝐺𝑟(𝑘,𝐸))=𝑃(𝐸)×𝑆𝐺𝑟(𝑘,𝐸).
For any morphism of S-schemes:

𝑇→𝐺𝑟(𝑘,𝐸),

this closed immersion induces a closed immersion

𝑃(𝐺𝑇)→𝑃(𝐸)×𝑆𝑇.

Conversely, any such closed immersion comes from a surjective homomorphism

of

𝑂𝑇

-modules from

𝐸𝑇

to a locally free module of rank

.[5] Therefore, the elements of

𝐺𝑟(𝑘,𝐸)(𝑇)

are exactly the projective subbundles of rank

in

𝑃(𝐸)×𝑆𝑇.
Under this identification, when

𝑇=𝑆

is the spectrum of a field

and

is given by a vector space

, the set of rational points

𝐺𝑟(𝑘,𝐸)(𝐾)

correspond to the projective linear subspaces of dimension

𝑘−1

in

𝑃(𝑉)

, and the image of

𝑃(𝐺)(𝐾)

in

𝑃(𝑉)×𝐾𝐺𝑟(𝑘,𝐸)
is the set

{(𝑥,𝑣)∈𝑃(𝑉)(𝐾)×𝐺𝑟(𝑘,𝐸)(𝐾)∣𝑥∈𝑣}.

The Plücker embedding[edit]

Main article: Plücker embedding

The Plücker embedding[6] is a natural embedding of the Grassmannian

𝐺𝑟(𝑘,𝑉)

into the projectivization of the

th Exterior power

Λ𝑘𝑉

of

𝜄:𝐺𝑟(𝑘,𝑉)→𝑃(Λ𝑘𝑉).

Suppose that
𝑤⊂𝑉

is a

-dimensional subspace of the

-dimensional vector space

. To define

𝜄(𝑤)

, choose a basis

(𝑤1,⋯,𝑤𝑘)

for

, and let

𝜄(𝑤)

be the projectivization of the wedge product of these basis elements:

𝜄(𝑤)=[𝑤1∧⋯∧𝑤𝑘],

where

[⋅]
 denotes the projective equivalence class.

A different basis for

will give a different wedge product, but the two will differ only by a non-zero
scalar multiple (the determinant of the change of basis matrix). Since the right-
hand side takes values in the projectivized space,

is well-defined. To see that it is an embedding, notice that it is possible to

recover

from

𝜄(𝑤)

as the span of the set of all vectors

𝑣∈𝑉

such that

𝑣∧𝜄(𝑤)=0

Plücker coordinates and Plücker relations[edit]

The Plücker embedding of the Grassmannian satisfies a set of simple quadratic

relations called the Plücker relations. These show that the Grassmannian

𝐺𝑟𝑘(𝑉)
 embeds as a nonsingular projective algebraic subvariety of the
projectivization

𝑃(Λ𝑘𝑉)

of the

th exterior power of

and give another method for constructing the Grassmannian. To state the
Plücker relations, fix a basis

(𝑒1,⋯,𝑒𝑛)

for

, and let

𝑤⊂𝑉

be a

-dimensional subspace of

with basis

(𝑤1,⋯,𝑤𝑘)
 . Let

(𝑤𝑖1,⋯,𝑤𝑖𝑛)

be the components of

𝑤𝑖

with respect to the chosen basis of

, and

(𝑊1,…,𝑊𝑛)

the

-component column vectors forming the transpose of the corresponding

homogeneous coordinate matrix:

𝑊𝑇=[𝑊1⋯𝑊𝑛]=[𝑤11⋯𝑤1𝑛⋮⋱⋮𝑤𝑘1⋯𝑤𝑘𝑛],

For any ordered sequence

1≤𝑖1<⋯<𝑖𝑘≤𝑛

of

𝑘
 positive integers, let

𝑤𝑖1,…,𝑖𝑘

be the determinant of the

𝑘×𝑘

matrix with columns

[𝑊𝑖1,…,𝑊𝑖𝑘]

. The elements

{𝑤𝑖1,…,𝑖𝑘|1≤𝑖1<⋯<𝑖𝑘≤𝑛}

are called the Plücker coordinates of the

element

𝑤∈𝐺𝑟𝑘(𝑉)

of the Grassmannian (with respect to the basis

(𝑒1,⋯,𝑒𝑛)

of

). These are the linear coordinates of the image

𝜄(𝑤)

of

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

Λ𝑘𝑉

space generated by the basis

(𝑒1,⋯,𝑒𝑛)

of

. Since a change of basis for

gives rise to multiplication of the Plücker coordinates by a nonzero constant

(the determinant of the change of basis matrix), these are only defined up to
projective equivalence, and hence determine a point in

𝑃(Λ𝑘𝑉)

For any two ordered sequences

1≤𝑖1<𝑖2⋯<𝑖𝑘−1≤𝑛

and

1≤𝑗1<𝑗2⋯<𝑗𝑘+1≤𝑛

of

𝑘−1

and
𝑘+1

positive integers, respectively, the following homogeneous quadratic

equations, known as the Plücker relations, or the Plücker-Grassmann relations,
are valid and determine the image

𝜄(𝐺𝑟𝑘(𝑉))

of

𝐺𝑟𝑘(𝑉)

under the Plücker map embedding:

∑𝑙=1𝑘+1(−1)ℓ𝑤𝑖1,…,𝑖𝑘−1,𝑗𝑙𝑤𝑗1,…,𝑗𝑙^,…𝑗𝑘+1=0,

where

𝑗1,…,𝑗𝑙^,…𝑗𝑘+1

denotes the sequence

𝑗1,…,𝑗𝑘+1

with the term

𝑗𝑙

omitted. These are consistent, determining a nonsingular projective algebraic

variety, but they are not algebraically independent. They are equivalent to the
statement that

𝜄(𝑤)
 is the projectivization of a completely decomposable element of

Λ𝑘𝑉

When

dim⁡(𝑉)=4

, and

𝑘=2

(the simplest Grassmannian that is not a projective space), the above

reduces to a single equation. Denoting the homogeneous coordinates of the
image

𝜄(𝐺𝑟2(𝑉)⊂𝑃(Λ2𝑉)

under the Plücker map as

(𝑤12,𝑤13,𝑤14,𝑤23,𝑤24,𝑤34)

, this single Plücker relation is

𝑤12𝑤34−𝑤13𝑤24+𝑤14𝑤23=0.

In general, many more equations are needed to define the image

𝜄(𝐺𝑟𝑘(𝑉))

of the Grassmannian in

𝑃(Λ𝑘𝑉)
 under the Plücker embedding.

Duality[edit]

Every

-dimensional subspace

𝑊⊂𝑉

determines an

(𝑛−𝑘)

-dimensional quotient space

𝑉/𝑊

of

. This gives the natural short exact sequence:

0→𝑊→𝑉→𝑉/𝑊→0.

Taking the dual to each of these three spaces and the dual linear transformations
yields an inclusion of

(𝑉/𝑊)∗

in
𝑉∗

with quotient

𝑊∗

0→(𝑉/𝑊)∗→𝑉∗→𝑊∗→0.

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

-dimensional subspaces of

and

(𝑛−𝑘)

-dimensional subspaces of

𝑉∗

. In terms of the Grassmannian, this gives a canonical isomorphism

𝐺𝑟𝑘(𝑉)↔𝐺𝑟(𝑛−𝑘,𝑉∗)

that associates to each subspace

𝑊⊂𝑉

its annihilator

𝑊0⊂𝑉∗

. Choosing an isomorphism of

with

𝑉∗

therefore determines a (non-canonical) isomorphism between

𝐺𝑟𝑘(𝑉)

and

𝐺𝑟𝑛−𝑘(𝑉)

. An isomorphism of

with

𝑉∗

is equivalent to the choice of an inner product, so with respect to the chosen

inner product, this isomorphism of Grassmannians sends any

-dimensional subspace into its

(𝑛−𝑘)
 }-dimensional orthogonal complement.

Schubert cells[edit]

The detailed study of Grassmannians makes use of a decomposition into affine

subpaces called Schubert cells, which were first applied in enumerative
geometry. The Schubert cells for

𝐺𝑟𝑘(𝑉)

are defined in terms of a specified complete flag of subspaces

𝑉1⊂𝑉2⊂⋯⊂𝑉𝑛=𝑉

of dimension

dim(𝑉𝑖)=𝑖

. For any integer partition

𝜆=(𝜆1,⋯,𝜆𝑘)

of weight

|𝜆|=∑𝑖=1𝑘𝜆𝑖

consisting of weakly decreasing non-negative integers

𝜆1≥⋯≥𝜆𝑘≥0,
whose Young diagram fits within the rectangular one

(𝑛−𝑘)𝑘

, the Schubert cell

𝑋𝜆(𝑘,𝑛)⊂𝐺𝑟𝑘(𝑉)

consists of those elements

𝑊∈𝐺𝑟𝑘(𝑉)

whose intersections with the subspaces

{𝑉𝑖}

have the following dimensions

𝑋𝜆(𝑘,𝑛)={𝑊∈𝐺𝑟𝑘(𝑉)|dim⁡(𝑊∩𝑉𝑛−𝑘+𝑗−𝜆𝑗)=𝑗}.

These are affine spaces, and their closures (within the Zariski topology) are
known as Schubert varieties.

As an example of the technique, consider the problem of determining the Euler

characteristic

𝜒𝑘,𝑛

of the Grassmannian

𝐺𝑟𝑘(𝑅𝑛)

n
of k-dimensional subspaces of R . Fix a

1
 -dimensional subspace

𝑅⊂𝑅𝑛

and consider the partition of

𝐺𝑟𝑘(𝑅𝑛)

n
into those k-dimensional subspaces of R that contain R and those that
do not. The former is

𝐺𝑟𝑘−1(𝑅𝑛−1)

and the latter is a rank

vector bundle over

𝐺𝑟𝑘(𝑅𝑛−1)

. This gives recursive formulae:

𝜒𝑘,𝑛=𝜒𝑘−1,𝑛−1+(−1)𝑘𝜒𝑘,𝑛−1,𝜒0,𝑛=𝜒𝑛,𝑛=1.

Solving these recursion relations gives the formula:

𝜒𝑘,𝑛=0

if

is even and
𝑘

is odd and

𝜒𝑘,𝑛=(⌊𝑛2⌋⌊𝑘2⌋)

otherwise.

Cohomology ring of the complex Grassmannian[edit]

Every point in the complex Grassmann manifold

𝐺𝑟𝑘(𝐶𝑛)

defines a

-plane in

-space. Fibering these planes over the Grassmannian one arrives at the vector
bundle

which generalizes the tautological bundle of a projective space. Similarly the

(𝑛−𝑘)

-dimensional orthogonal complements of these planes yield an

orthogonal vector bundle
𝐹

. The integral cohomology of the Grassmannians is generated, as a ring, by the

Chern classes of

. 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

and

. Then the relations merely state that the direct sum of the bundles

and

is trivial. Functoriality of the total Chern classes allows one to write this
relation as

𝑐(𝐸)𝑐(𝐹)=1.

The quantum cohomology ring was calculated by Edward Witten.[7] The

generators are identical to those of the classical cohomology ring, but the top
relation is changed to

𝑐𝑘(𝐸)𝑐𝑛−𝑘(𝐹)=(−1)𝑛−𝑘
reflecting the existence in the corresponding quantum field theory of an instanton
with

2𝑛

fermionic zero-modes which violates the degree of the cohomology

corresponding to a state by

2𝑛

units.

Associated measure[edit]

When

is an

-dimensional Euclidean space, we may define a uniform measure on

𝐺𝑟𝑘(𝑉)

in the following way. Let

𝜃𝑛

be the unit Haar measure on the orthogonal group

𝑂(𝑛)
 and fix

𝑤∈𝐺𝑟𝑘(𝑉)

. Then for a set

𝐴⊂𝐺𝑟𝑘(𝑉)

, define

𝛾𝑘,𝑛(𝐴)=𝜃𝑛{𝑔∈O⁡(𝑛):𝑔𝑤∈𝐴}.

This measure is invariant under the action of the group

𝑂(𝑛)

; that is,

𝛾𝑘,𝑛(𝑔𝐴)=𝛾𝑘,𝑛(𝐴)

for all

𝑔∈𝑂(𝑛)

. Since

𝜃𝑛(𝑂(𝑛))=1

, we have

𝛾𝑘,𝑛(𝐺𝑟𝑘(𝑉))=1
 . Moreover,

𝛾𝑘,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[edit]

This is the manifold consisting of all oriented

-dimensional subspaces of

𝑅𝑛

. It is a double cover of

𝐺𝑟𝑘(𝑅𝑛)

and is denoted by

𝐺𝑟~𝑘(𝑅𝑛)

As a homogeneous space it can be expressed as:

𝐺𝑟~𝑘(𝑅𝑛)=SO⁡(𝑛)/(SO⁡(𝑘)×SO⁡(𝑛−𝑘)).

Orthogonal isotropic Grassmannians[edit]

Given a real or complex nondegenerate symmetric bilinear form

on the

-dimensional space

(i.e., a scalar product), the totally isotropic Grassmannian

𝐺𝑟𝑘0(𝑉,𝑄)

is defined as the subvariety

𝐺𝑟𝑘0(𝑉,𝑄)⊂𝐺𝑟𝑘(𝑉)

consisting of all

-dimensional subspaces

𝑤⊂𝑉

for which

𝑄(𝑢,𝑣)=0,∀𝑢,𝑣∈𝑤.

Maximal isotropic Grassmannians with respect to a real or complex scalar

product are closely related to Cartan's theory of spinors.[8] Under the Cartan
embedding, their connected components are equivariantly diffeomorphic to the
projectivized minimal spinor orbit, under the spin representation, the so-called
projective pure spinor variety which, similarly to the image of the Plücker map
embedding, is cut out as the intersection of a number of quadrics, the Cartan
quadrics.[8][9][10]

Applications[edit]

A key application of Grassmannians is as the "universal" embedding space for

bundles with connections on compact manifolds. [11][12]

Another important application is Schubert calculus, which is the enumerative

geometry involved in calculating the number of points, lines, planes, etc. in a
projective space that intersect a given set of points, lines, etc., using the
intersection theory of Schubert varieties. Subvarieties of Schubert cells can also
be used to parametrize simultaneous eigenvectors of complete sets of
commuting operators in quantum integrable spin systems, such as the Gaudin
model, using the Bethe ansatz method.[13]

A further application is to the solution of hierarchies of classical completely

integrable systems of partial differential equations, such as the Kadomtsev–
Petviashvili equation and the associated KP hierarchy. These can be expressed in
terms of abelian group flows on an infinite-dimensional Grassmann manifold. [14]
[15][16][17]
The KP equations, expressed in Hirota bilinear form in terms of the KP
Tau function are equivalent to the Plücker relations.[18][17] A similar construction
holds for solutions of the BKP integrable hierarchy, in terms of abelian group
flows on an infinite dimensional maximal isotropic Grassmann manifold. [15][16][19]

Finite dimensional positive Grassmann manifolds can be used to express soliton

solutions of KP equations which are nonsingular for real values of the KP flow
parameters.[20][21][22]

The scattering amplitudes of subatomic particles in maximally supersymmetric

super Yang-Mills theory may be calculated in the planar limit via a positive
Grassmannian construct called the amplituhedron.[23]

Grassmann manifolds have also found applications in computer vision tasks of

video-based face recognition and shape recognition, [24] and are used in the data-
visualization technique known as the grand tour.
See also[edit]

● Schubert calculus
● For an example of the use of Grassmannians in differential geometry,
see Gauss map
● In projective geometry, see Plücker embedding and Plücker co-
ordinates.
● Flag manifolds are generalizations of Grassmannians whose elements,
viewed geometrically, are nested sequences of subspaces of specified
dimensions.
● Stiefel manifolds are bundles of orthonormal frames over
Grassmanians.
● Given a distinguished class of subspaces, one can define
Grassmannians of these subspaces, such as Isotropic Grassmanians or
Lagrangian Grassmannians .
● Isotropic Grassmanian
● Lagrangian Grassmannian
● Grassmannians provide classifying spaces in K-theory, notably the
classifying space for U(n). In the homotopy theory of schemes, the
Grassmannian plays a similar role for algebraic K-theory.[25]
● Affine Grassmannian
● Grassmann bundle
● Grassmann graph

Notes[edit]

1. ^ Lee 2012, p. 22, Example 1.36.

2. ^ Shafarevich 2013, p. 42, Example 1.24.
3. ^ Milnor & Stasheff (1974), pp. 57–59.
4. ^ Grothendieck, Alexander (1971). Éléments de géométrie algébrique. Vol. 1 (2nd ed.).
Berlin, New York: Springer-Verlag. ISBN 978-3-540-05113-8., Chapter I.9
5. ^ EGA, II.3.6.3.
6. ^ Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley
Classics Library (2nd ed.), New York: John Wiley & Sons, p. 211, ISBN 0-471-05059-8,
MR 1288523, Zbl 0836.14001
7. ^ Witten, Edward (1993). "The Verlinde algebra and the cohomology of the
Grassmannian". arXiv:hep-th/9312104.
8. ^
9. Jump up to:
ab
10. Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications.
ISBN 978-0-486-64070-9. MR 0631850.
11. ^ Harnad, J.; Shnider, S. (1992). "Isotropic geometry and twistors in higher
dimensions. I. The generalized Klein correspondence and spinor flags in even
dimensions". Journal of Mathematical Physics. 33 (9). American Institute of Physics:
3197–3208. Bibcode:1992JMP....33.3197H. doi:10.1063/1.529538.
12. ^ Harnad, J.; Shnider, S. (1995). "Isotropic geometry and twistors in higher
dimensions. II. Odd dimensions, reality conditions, and twistor superspaces". Journal
of Mathematical Physics. 36 (9). American Institute of Physics: 1945–1970.
Bibcode:1995JMP....36.1945H. doi:10.1063/1.531096.
13. ^ Narasimhan, M. S.; Ramanan, S. (1961). "Existence of Universal Connections".
American Journal of Mathematics. 83 (3): 563–572. doi:10.2307/2372896.
hdl:10338.dmlcz/700905. JSTOR 2372896. S2CID 123324468.
14. ^ Narasimhan, M. S.; Ramanan, S. (1963). "Existence of Universal Connections II".
American Journal of Mathematics. 85 (2): 223–231. doi:10.2307/2373211. JSTOR
2373211.
15. ^ Mukhin, E.; Tarasov, V.; Varchenko, A. (2009). "Schubert Calculus and
representations of the general linear group". J. Amer. Math. Soc. 22 (4). American
Mathematical Society: 909–940. arXiv:0711.4079. doi:10.1090/S0894-0347-09-00640-7.
16. ^ M. Sato, "Soliton equations as dynamical systems on infinite dimensional
Grassmann manifolds", Kokyuroku, RIMS, Kyoto Univ., 30–46 (1981).
17. ^
18. Jump up to:
ab
19. Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator
Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton
Equations III–". Journal of the Physical Society of Japan. 50 (11). Physical Society of
Japan: 3806–3812. Bibcode:1981JPSJ...50.3806D. doi:10.1143/jpsj.50.3806. ISSN 0031-
9015.
20. ^
21. Jump up to:
ab
22. Jimbo, Michio; Miwa, Tetsuji (1983). "Solitons and infinite-dimensional Lie
algebras". Publications of the Research Institute for Mathematical Sciences. 19 (3).
European Mathematical Society Publishing House: 943–1001.
doi:10.2977/prims/1195182017. ISSN 0034-5318.
23. ^
24. Jump up to:
ab
25. Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapts. 4 and 5.
Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge
University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
26. ^ Sato, Mikio (October 1981). "Soliton Equations as Dynamical Systems on Infinite
Dimensional Grassmann Manifolds (Random Systems and Dynamical Systems)". 数理
解析研究所講究録. 439: 30–46. hdl:2433/102800.
27. ^ Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapt. 7.
Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge
University Press. doi:10.1017/9781108610902. ISBN 9781108610902. S2CID 222379146.
28. ^ Chakravarty, S.; Kodama, Y. (July 2009). "Soliton Solutions of the KP Equation and
Application to Shallow Water Waves". Studies in Applied Mathematics. 123: 83–151.
arXiv:0902.4433. doi:10.1111/j.1467-9590.2009.00448.x. S2CID 18390193.
29. ^ Kodama, Yuji; Williams, Lauren (December 2014). "KP solitons and total positivity for
the Grassmannian". Inventiones Mathematicae. 198 (3): 637–699. arXiv:1106.0023.
Bibcode:2014InMat.198..637K. doi:10.1007/s00222-014-0506-3. S2CID 51759294.
30. ^ Hartnett, Kevin (16 December 2020). "A Mathematician's Unanticipated Journey
Through the Physical World". Quanta Magazine. Retrieved 17 December 2020.
31. ^ Arkani-Hamed, Nima; Trnka, Jaroslav (2013). "The Amplituhedron". Journal of High
Energy Physics. 2014 (10): 30. arXiv:1312.2007. Bibcode:2014JHEP...10..030A.
doi:10.1007/JHEP10(2014)030. S2CID 7717260.
32. ^ Pavan Turaga, Ashok Veeraraghavan, Rama Chellappa: Statistical analysis on Stiefel
and Grassmann manifolds with applications in computer vision, CVPR 23–28 June
2008, IEEE Conference on Computer Vision and Pattern Recognition, 2008, ISBN 978-1-
4244-2242-5, pp. 1–8 (abstract, full text)
1
33. ^ Morel, Fabien; Voevodsky, Vladimir (1999). "A -homotopy theory of schemes" (PDF).
Publications Mathématiques de l'IHÉS. 90 (90): 45–143. doi:10.1007/BF02698831. ISSN
 1618-1913. MR 1813224. S2CID 14420180. Retrieved 2008-09-05., see section 4.3., pp.
137–140

References[edit]

● Griffiths, Phillip; Harris, Joseph (1994). Principles of algebraic

geometry. Wiley Classics Library (2nd ed.). New York: John Wiley &
Sons. p. 211. ISBN 0-471-05059-8. MR 1288523. Zbl 0836.14001.
● Hatcher, Allen (2003). Vector Bundles & K-Theory (PDF) (2.0 ed.). section
1.2
● Milnor, John W.; Stasheff, James D. (1974). Characteristic classes.
Annals of Mathematics Studies. Vol. 76. Princeton, NJ: Princeton
University Press. ISBN 0-691-08122-0. see chapters 5–7
● Harris, Joe (1992). Algebraic Geometry: A First Course. New York:
Springer. ISBN 0-387-97716-3.
● Helgason, Sigurdur (1978), Differential geometry, Lie groups, and
symmetric spaces, Academic Press, ISBN 978-0-8218-2848-9
● Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts
in Mathematics. Vol. 218 (Second ed.). New York London: Springer-
Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.
● Mattila, Pertti (1995). Geometry of Sets and Measures in Euclidean
Spaces. New York: Cambridge University Press. ISBN 0-521-65595-1.
● Shafarevich, Igor R. (2013). Basic Algebraic Geometry 1. Springer
Science. doi:10.1007/978-3-642-37956-7. ISBN 978-0-387-97716-4.

Categories:

Differential geometry
Projective geometry
Algebraic homogeneous spaces
Algebraic geometry
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