Grassmannian

Short description: Mathematical space

In mathematics, the Grassmannian [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all [math]\displaystyle{ k }[/math]-dimensional linear subspaces of an [math]\displaystyle{ n }[/math]-dimensional vector space [math]\displaystyle{ V }[/math] over a field [math]\displaystyle{ K }[/math]. For example, the Grassmannian [math]\displaystyle{ \mathbf{Gr}_1(V) }[/math] is the space of lines through the origin in [math]\displaystyle{ V }[/math], so it is the same as the projective space [math]\displaystyle{ \mathbf{P}(V) }[/math] of one dimension lower than [math]\displaystyle{ V }[/math]. When [math]\displaystyle{ V }[/math] is a real or complex vector space, Grassmannians are compact smooth manifolds , of dimension [math]\displaystyle{ k(n-k) }[/math]. 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 [math]\displaystyle{ \mathbf{Gr}_2(\mathbf{R}^4) }[/math], 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 [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math], [math]\displaystyle{ \mathbf{Gr}(k,V) }[/math],[math]\displaystyle{ \mathbf{Gr}_k(n) }[/math], [math]\displaystyle{ \mathbf{Gr}(k,n) }[/math] to denote the Grassmannian of [math]\displaystyle{ k }[/math]-dimensional subspaces of an [math]\displaystyle{ n }[/math]-dimensional vector space [math]\displaystyle{ V }[/math].

Motivation

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 [math]\displaystyle{ M }[/math] of dimension [math]\displaystyle{ k }[/math] embedded in [math]\displaystyle{ \mathbf{R}^n }[/math]. At each point [math]\displaystyle{ x\in M }[/math], the tangent space to [math]\displaystyle{ M }[/math] can be considered as a subspace of the tangent space of [math]\displaystyle{ \mathbf{R}^n }[/math], which is also just [math]\displaystyle{ \mathbf{R}^n }[/math]. The map assigning to [math]\displaystyle{ x }[/math] its tangent space defines a map from M to [math]\displaystyle{ \mathbf{Gr}_k(\mathbf{R}^n) }[/math]. (In order to do this, we have to translate the tangent space at each [math]\displaystyle{ x \in M }[/math] so that it passes through the origin rather than [math]\displaystyle{ x }[/math], and hence defines a [math]\displaystyle{ k }[/math]-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 [math]\displaystyle{ M }[/math], so that every vector bundle generates a continuous map from [math]\displaystyle{ M }[/math] 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

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 [math]\displaystyle{ \mathbf{P}^{n-1} }[/math]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 that plane (and vice versa); hence the spaces 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 Gr(2, 4).

The Grassmannian as a differentiable manifold

To endow [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] with the structure of a differentiable manifold, choose a basis for [math]\displaystyle{ V }[/math]. This is equivalent to identifying [math]\displaystyle{ V }[/math] with [math]\displaystyle{ K^n }[/math], with the standard basis denoted [math]\displaystyle{ (e_1, \dots, e_n) }[/math], viewed as column vectors. Then for any [math]\displaystyle{ k }[/math]-dimensional subspace [math]\displaystyle{ w\subset V }[/math], viewed as an element of [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math], we may choose a basis consisting of [math]\displaystyle{ k }[/math] linearly independent column vectors [math]\displaystyle{ (W_1, \dots, W_k) }[/math]. The homogeneous coordinates of the element [math]\displaystyle{ w \in \mathbf{Gr}_k(V) }[/math] consist of the elements of the [math]\displaystyle{ n\times k }[/math] maximal rank rectangular matrix [math]\displaystyle{ W }[/math] whose [math]\displaystyle{ i }[/math]-th column vector is [math]\displaystyle{ W_i }[/math], [math]\displaystyle{ i = 1, \dots, k }[/math]. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices [math]\displaystyle{ W }[/math] and [math]\displaystyle{ \tilde{W} }[/math] represent the same element [math]\displaystyle{ w \in \mathbf{Gr}_k(V) }[/math] if and only if

[math]\displaystyle{ \tilde{W} = W g }[/math]

for some element [math]\displaystyle{ g \in GL(k, K) }[/math] of the general linear group of invertible [math]\displaystyle{ k\times k }[/math] matrices with entries in [math]\displaystyle{ K }[/math]. This defines an equivalence relation between [math]\displaystyle{ n\times k }[/math] matrices [math]\displaystyle{ W }[/math] of rank [math]\displaystyle{ k }[/math], for which the equivalence classes are denoted [math]\displaystyle{ [W] }[/math].

We now define a coordinate atlas. For any [math]\displaystyle{ n \times k }[/math] homogeneous coordinate matrix [math]\displaystyle{ W }[/math], we can apply elementary column operations (which amounts to multiplying [math]\displaystyle{ W }[/math] by a sequence of elements [math]\displaystyle{ g \in GL(k, K) }[/math]) to obtain its reduced column echelon form. If the first [math]\displaystyle{ k }[/math] rows of [math]\displaystyle{ W }[/math] are linearly independent, the result will have the form

[math]\displaystyle{ \begin{bmatrix} 1 \\ & 1 \\ & & \ddots \\ & & & 1 \\ a_{1,1} & \cdots & \cdots & a_{1,k} \\ \vdots & & & \vdots \\ a_{n-k,1} & \cdots & \cdots & a_{n-k,k} \end{bmatrix} }[/math]

and the [math]\displaystyle{ (n-k)\times k }[/math] affine coordinate matrix [math]\displaystyle{ A }[/math] with entries [math]\displaystyle{ (a_{ij}) }[/math] determines [math]\displaystyle{ w }[/math]. In general, the first [math]\displaystyle{ k }[/math] rows need not be independent, but since [math]\displaystyle{ W }[/math] has maximal rank [math]\displaystyle{ k }[/math], there exists an ordered set of integers [math]\displaystyle{ 1 \le i_1 \lt \cdots \lt i_k \le n }[/math] such that the [math]\displaystyle{ k \times k }[/math] submatrix [math]\displaystyle{ W_{i_1, \dots, i_k} }[/math] whose rows are the [math]\displaystyle{ (i_1, \ldots, i_k) }[/math]-th rows of [math]\displaystyle{ W }[/math] is nonsingular. We may apply column operations to reduce this submatrix to the identity matrix, and the remaining entries uniquely determine [math]\displaystyle{ w }[/math]. Hence we have the following definition:

For each ordered set of integers [math]\displaystyle{ 1 \le i_1 \lt \cdots \lt i_k \le n }[/math], let [math]\displaystyle{ U_{i_1, \dots, i_k} }[/math] be the set of elements [math]\displaystyle{ w\in \mathbf{Gr}_k(V) }[/math] for which, for any choice of homogeneous coordinate matrix [math]\displaystyle{ W }[/math], the [math]\displaystyle{ k\times k }[/math] submatrix [math]\displaystyle{ W_{i_1, \dots, i_k} }[/math] whose [math]\displaystyle{ j }[/math]-th row is the [math]\displaystyle{ i_j }[/math]-th row of [math]\displaystyle{ W }[/math] is nonsingular. The affine coordinate functions on [math]\displaystyle{ U_{i_1, \dots, i_k} }[/math] are then defined as the entries of the [math]\displaystyle{ (n-k)\times k }[/math] matrix [math]\displaystyle{ A^{i_1, \dots, i_k} }[/math] whose rows are those of the matrix [math]\displaystyle{ W W^{-1}_{i_1, \dots, i_k} }[/math] complementary to [math]\displaystyle{ (i_1, \dots, i_k) }[/math], written in the same order. The choice of homogeneous [math]\displaystyle{ n \times k }[/math] coordinate matrix [math]\displaystyle{ W }[/math] in [math]\displaystyle{ [W] }[/math] representing the element [math]\displaystyle{ w\in \mathbf{Gr}_k(V) }[/math] does not affect the values of the affine coordinate matrix [math]\displaystyle{ A^{i_1, \dots, i_k} }[/math] representing w on the coordinate neighbourhood [math]\displaystyle{ U_{i_1, \dots, i_k} }[/math]. Moreover, the coordinate matrices [math]\displaystyle{ A^{i_1, \dots, i_k} }[/math] may take arbitrary values, and they define a diffeomorphism from [math]\displaystyle{ U_{i_1, \dots, i_k} }[/math] to the space of [math]\displaystyle{ K }[/math]-valued [math]\displaystyle{ (n-k)\times k }[/math] matrices. Denote by

[math]\displaystyle{ \hat{A}^{i_1, \dots, i_k} := W (W_{i_1, \dots , i_k})^{-1} }[/math]

the homogeneous coordinate matrix having the identity matrix as the [math]\displaystyle{ k \times k }[/math] submatrix with rows [math]\displaystyle{ (i_1, \dots, i_k) }[/math] and the affine coordinate matrix [math]\displaystyle{ A^{i_1, \dots, i_k} }[/math] in the consecutive complementary rows. On the overlap [math]\displaystyle{ U_{i_1, \dots, i_k} \cap U_{j_1, \dots, j_k} }[/math] between any two such coordinate neighborhoods, the affine coordinate matrix values [math]\displaystyle{ A^{i_1, \dots, i_k} }[/math] and [math]\displaystyle{ A^{j_1, \dots, j_k} }[/math] are related by the transition relations

[math]\displaystyle{ \hat{A}^{i_1, \dots, i_k} W_{i_1, \dots, i_k} = \hat{A}^{j_1, \dots, j_k} W_{j_1, \dots, j_k}, }[/math]

where both [math]\displaystyle{ W_{i_1, \dots, i_k} }[/math] and [math]\displaystyle{ W_{j_1, \dots, j_k} }[/math] are invertible. This may equivalently be written as

[math]\displaystyle{ \hat{A}^{j_1, \dots, j_k} = \hat{A}^{i_1, \dots, i_k} (\hat{A}^{i_1, \dots, i_k}_{j_1, \dots, j_k})^{-1}, }[/math]

where [math]\displaystyle{ \hat{A}^{i_1, \dots, i_k}_{j_1, \dots, j_k} }[/math] is the invertible [math]\displaystyle{ k \times k }[/math] matrix whose [math]\displaystyle{ l }[/math]th row is the [math]\displaystyle{ j_l }[/math]th row of [math]\displaystyle{ \hat{A}^{i_1, \dots, i_k} }[/math]. The transition functions are therefore rational in the matrix elements of [math]\displaystyle{ A^{i_1, \dots, i_k} }[/math], and [math]\displaystyle{ \{U_{i_1, \dots, i_k}, A^{i_1, \dots, i_k}\} }[/math] gives an atlas for [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] as a differentiable manifold and also as an algebraic variety.

The Grassmannian as a set of orthogonal projections

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) problem 5-C). For this, choose a positive definite real or Hermitian inner product [math]\displaystyle{ \langle \cdot , \cdot \rangle }[/math] on [math]\displaystyle{ V }[/math], depending on whether [math]\displaystyle{ V }[/math] is real or complex. A [math]\displaystyle{ k }[/math]-dimensional subspace [math]\displaystyle{ w }[/math] determines a unique orthogonal projection operator [math]\displaystyle{ P_w:V\rightarrow V }[/math] whose image is [math]\displaystyle{ w\subset V }[/math] by splitting [math]\displaystyle{ V }[/math] into the orthogonal direct sum

[math]\displaystyle{ V = w \oplus w^\perp }[/math]

of [math]\displaystyle{ w }[/math] and its orthogonal complement [math]\displaystyle{ w^\perp }[/math] and defining

[math]\displaystyle{ P_w(v) =\begin{cases} v \quad \text{ if } v \in w \\ 0 \quad \text{ if } v\in w^\perp . \end{cases} }[/math]

Conversely, every projection operator [math]\displaystyle{ P }[/math] of rank [math]\displaystyle{ k }[/math] defines a subspace [math]\displaystyle{ w_P := \mathrm{Im}(P) }[/math] as its image. Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold [math]\displaystyle{ \mathbf{Gr}(k, V) }[/math] with the set of rank [math]\displaystyle{ k }[/math] orthogonal projection operators [math]\displaystyle{ P }[/math]:

[math]\displaystyle{ \mathbf{Gr}(k, V) \sim \left\{ P \in \mathrm{End}(V) \mid P = P^2 = P^\dagger,\, \mathrm{tr}(P) = k \right\}. }[/math]

In particular, taking [math]\displaystyle{ V = \mathbf{R}^n }[/math] or [math]\displaystyle{ V = \mathbf{C}^n }[/math] this gives completely explicit equations for embedding the Grassmannians [math]\displaystyle{ \mathbf{Gr}(k, \mathbf{R}^N) }[/math], [math]\displaystyle{ \mathbf{Gr}(k, \mathbf{C}^N) }[/math] in the space of real or complex [math]\displaystyle{ n\times n }[/math] matrices [math]\displaystyle{ \mathbf{R}^{n \times n} }[/math], [math]\displaystyle{ \mathbf{C}^{n \times n} }[/math], respectively.

Since this defines the Grassmannian as a closed subset of the sphere [math]\displaystyle{ \{X \in \mathrm{End}(V) \mid \mathrm{tr}(XX^\dagger) = k\} }[/math] this is one way to see that the Grassmannian is a compact Hausdorff space. This construction also turns the Grassmannian [math]\displaystyle{ \mathbf{Gr}(k, V) }[/math] into a metric space with metric

[math]\displaystyle{ d(w, w') := \lVert P_w - P_{w'} \rVert, }[/math]

for any pair [math]\displaystyle{ w, w' \subset V }[/math] of [math]\displaystyle{ k }[/math]-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 [math]\displaystyle{ V }[/math], 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 [math]\displaystyle{ \mathbf{Gr}(k, \mathbf{R}^n) }[/math], [math]\displaystyle{ \mathbf{Gr}(k, \mathbf{C}^n) }[/math] as affine algebraic varieties

Let [math]\displaystyle{ M(n, \mathbf{R}) }[/math] denote the space of real [math]\displaystyle{ n \times n }[/math] matrices and the subset [math]\displaystyle{ P(k, n, \mathbf{R})\subset M(n, \mathbf{R}) }[/math] of matrices [math]\displaystyle{ P \in M(n, \mathbf{R}) }[/math] that satisfy the three conditions:

• [math]\displaystyle{ P }[/math] is a projection operator: [math]\displaystyle{ P^2=P }[/math].
• [math]\displaystyle{ P }[/math] is symmetric: [math]\displaystyle{ P^T=P }[/math].
• [math]\displaystyle{ P }[/math] has trace [math]\displaystyle{ \operatorname{tr}(P)=k }[/math].

There is a bijective correspondence between [math]\displaystyle{ P(k, n, \mathbf{R}) }[/math] and the Grassmannian [math]\displaystyle{ \mathbf{Gr}(k, \mathbf{R}^n) }[/math] of [math]\displaystyle{ k }[/math]-dimensional subspaces of [math]\displaystyle{ \mathbf{R}^n }[/math] given by sending [math]\displaystyle{ P\in P(k, n, \mathbf{R}) }[/math] to the [math]\displaystyle{ k }[/math]-dimensional subspace of [math]\displaystyle{ \mathbf{R}^n }[/math] spanned by its columns and, conversely, sending any element [math]\displaystyle{ w\in\mathbf{Gr}(k, \mathbf{R}^n) }[/math] to the projection matrix

[math]\displaystyle{ P_w:= \sum_{i=1}^k w_i w_i^T, }[/math]

where [math]\displaystyle{ (w_1, \cdots, w_k) }[/math] is any orthonormal basis for [math]\displaystyle{ w\subset\mathbf{R}^n }[/math], viewed as real [math]\displaystyle{ n }[/math] component column vectors.

An analogous construction applies to the complex Grassmannian [math]\displaystyle{ \mathbf{Gr}(k, \mathbf{C}^n) }[/math], identifying it bijectively with the subset [math]\displaystyle{ P(k, n, \mathbf{C})\subset M(n,\mathbf{C}) }[/math] of complex [math]\displaystyle{ n \times n }[/math] matrices [math]\displaystyle{ P\in M(n,\mathbf{C}) }[/math] satisfying

• [math]\displaystyle{ P }[/math] is a projection operator: [math]\displaystyle{ P^2=P }[/math].
• [math]\displaystyle{ P }[/math] is self-adjoint (Hermitian): [math]\displaystyle{ P^\dagger=P }[/math].
• [math]\displaystyle{ P }[/math] has trace [math]\displaystyle{ \rm{tr}(P)=k }[/math],

where the self-adjointness is with respect to the Hermitian inner product [math]\displaystyle{ \langle \, \cdot, \cdot \, \rangle }[/math] in which the standard basis vectors [math]\displaystyle{ (e_1, \cdots, e_n) }[/math] are orthonomal. The formula for the orthogonal projection matrix [math]\displaystyle{ P_w }[/math] onto the complex [math]\displaystyle{ k }[/math]-dimensional subspace [math]\displaystyle{ w\subset \mathbf{C}^n }[/math] spanned by the orthonormal (unitary) basis vectors [math]\displaystyle{ (w_1, \cdots, w_k) }[/math] is

[math]\displaystyle{ P_w:= \sum_{i=1}^k w_i w_i^\dagger. }[/math]

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]\displaystyle{ \mathrm{GL}(V) }[/math] acts transitively on the [math]\displaystyle{ k }[/math]-dimensional subspaces of [math]\displaystyle{ V }[/math]. Therefore, if we choose a subspace [math]\displaystyle{ w_0 \subset V }[/math] of dimension [math]\displaystyle{ k }[/math], any element [math]\displaystyle{ w\in\mathbf{Gr}(k, V) }[/math] can be expressed as

[math]\displaystyle{ w = g (w_0) }[/math]

for some group element [math]\displaystyle{ g \in \mathrm{GL}(V) }[/math], where [math]\displaystyle{ g }[/math] is determined only up to right multiplication by elements [math]\displaystyle{ \{h \in H\} }[/math] of the stabilizer of [math]\displaystyle{ w_0 }[/math]:

[math]\displaystyle{ H:=\mathrm{stab}(w_0):=\{h\in \mathrm{GL}(V) \,|\, h(w_0)=w_0 \} \subset \mathrm{GL}(V) }[/math]

under the [math]\displaystyle{ \mathrm{GL}(V) }[/math]-action.

We may therefore identify [math]\displaystyle{ \mathbf{Gr}(k, V) }[/math] with the quotient space

[math]\displaystyle{ \mathbf{Gr}(k, V) = \mathrm{GL}(V)/H }[/math]

of left cosets of [math]\displaystyle{ H }[/math].

If the underlying field is [math]\displaystyle{ \mathbf{R} }[/math] or [math]\displaystyle{ \mathbf{C} }[/math] and [math]\displaystyle{ \mathrm{GL}(V) }[/math] is considered as a Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a ground field [math]\displaystyle{ K }[/math], the group [math]\displaystyle{ \mathrm{GL}(V) }[/math] 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, [math]\displaystyle{ H }[/math] is a parabolic subgroup of [math]\displaystyle{ \mathrm{GL}(V) }[/math].

Over [math]\displaystyle{ \mathbf{R} }[/math] or [math]\displaystyle{ \mathbf{C} }[/math] it also becomes possible to use smaller groups in this construction. To do this over [math]\displaystyle{ \mathbf{R} }[/math], fix a Euclidean inner product [math]\displaystyle{ q }[/math] on [math]\displaystyle{ V }[/math]. The real orthogonal group [math]\displaystyle{ O(V, q) }[/math] acts transitively on the set of [math]\displaystyle{ k }[/math]-dimensional subspaces [math]\displaystyle{ \mathbf{Gr}(k, V) }[/math] and the stabiliser of a [math]\displaystyle{ k }[/math]-space [math]\displaystyle{ w_0\subset V }[/math] is

[math]\displaystyle{ O(w_0, q|_{w_0})\times O(w^\perp_0, q|_{w^\perp_0}) }[/math],

where [math]\displaystyle{ w_0^\perp }[/math] is the orthogonal complement of [math]\displaystyle{ w_0 }[/math] in [math]\displaystyle{ V }[/math]. This gives an identification as the homogeneous space

[math]\displaystyle{ \mathbf{Gr}(k, V) = O(V, q)/\left(O(w, q|_w)\times O(w^\perp, q|_{w^\perp})\right) }[/math].

If we take [math]\displaystyle{ V = \mathbf{R}^n }[/math] and [math]\displaystyle{ w_0 = \mathbf{R}^k \subset \mathbf{R}^n }[/math] (the first [math]\displaystyle{ k }[/math] components) we get the isomorphism

[math]\displaystyle{ \mathbf{Gr}(k,\mathbf{R}^n) = O(n)/\left(O(k) \times O(n - k)\right). }[/math]

Over C, if we choose an Hermitian inner product [math]\displaystyle{ h }[/math], the unitary group [math]\displaystyle{ U(V, h) }[/math] acts transitively, and we find analogously

[math]\displaystyle{ \mathbf{Gr}(k, V) = U(V , h)/\left(U(w_0, h|_{w_0}) \times U(w_0^\perp|, h_{w_0^\perp})\right), }[/math]

or, for [math]\displaystyle{ V = \mathbf{C}^n }[/math] and [math]\displaystyle{ w_0 = \mathbf{C}^k \subset \mathbf{C}^n }[/math],

[math]\displaystyle{ \mathbf{Gr}(k, \mathbf{C}^n) = U(n)/\left(U(k) \times U(n-k)\right). }[/math]

In particular, this shows that the Grassmannian is compact, and of (real or complex) dimension k(n − k).

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.

Representable functor

Let [math]\displaystyle{ \mathcal E }[/math] be a quasi-coherent sheaf on a scheme [math]\displaystyle{ S }[/math]. Fix a positive integer [math]\displaystyle{ k }[/math]. Then to each [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ T }[/math], the Grassmannian functor associates the set of quotient modules of

[math]\displaystyle{ \mathcal{E}_T := \mathcal E \otimes_{O_S} O_T }[/math]

locally free of rank [math]\displaystyle{ k }[/math] on [math]\displaystyle{ T }[/math]. We denote this set by [math]\displaystyle{ \mathbf{Gr}(k, \mathcal{E}_T) }[/math].

This functor is representable by a separated [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ \mathbf{Gr}(k, \mathcal{E}) }[/math]. The latter is projective if [math]\displaystyle{ \mathcal {E} }[/math] is finitely generated. When [math]\displaystyle{ S }[/math] is the spectrum of a field [math]\displaystyle{ K }[/math], then the sheaf [math]\displaystyle{ \mathcal{E} }[/math] is given by a vector space [math]\displaystyle{ V }[/math] and we recover the usual Grassmannian variety of the dual space of [math]\displaystyle{ V }[/math], namely: [math]\displaystyle{ \mathbf{Gr}(k, V) }[/math]. By construction, the Grassmannian scheme is compatible with base changes: for any [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ S' }[/math], we have a canonical isomorphism

[math]\displaystyle{ \mathbf{Gr}(k, \mathcal{E} ) \times_S S' \simeq \mathbf{Gr}(k, \mathcal{E}_{S'}) }[/math]

In particular, for any point [math]\displaystyle{ s }[/math] of [math]\displaystyle{ S }[/math], the canonical morphism [math]\displaystyle{ \{s\} = \text{Spec}K(s) \rightarrow S }[/math] induces an isomorphism from the fiber [math]\displaystyle{ \mathbf{Gr}(k, \mathcal {E})_s }[/math] to the usual Grassmannian [math]\displaystyle{ \mathbf{Gr}(k, \mathcal{E} \otimes_{O_S} K(s)) }[/math] over the residue field [math]\displaystyle{ K(s) }[/math].

Universal family

Since the Grassmannian scheme represents a functor, it comes with a universal object, [math]\displaystyle{ \mathcal G }[/math], which is an object of [math]\displaystyle{ \mathbf{Gr} \left (k, \mathcal{E}_{\mathbf {Gr}(k, \mathcal E)} \right), }[/math] and therefore a quotient module [math]\displaystyle{ \mathcal G }[/math] of [math]\displaystyle{ \mathcal E_{\mathbf {Gr}(k, \mathcal E)} }[/math], locally free of rank [math]\displaystyle{ k }[/math] over [math]\displaystyle{ \mathbf{Gr}(k, \mathcal{E}) }[/math]. The quotient homomorphism induces a closed immersion from the projective bundle:

[math]\displaystyle{ \mathbf{P}(\mathcal G) \to \mathbf{P} \left (\mathcal E_{\mathbf{Gr}(k, \mathcal E)} \right) = \mathbf P({\mathcal E}) \times_S \mathbf{Gr}(k, \mathcal E). }[/math]

For any morphism of S-schemes:

[math]\displaystyle{ T \to \mathbf{Gr}(k, \mathcal{E}), }[/math]

this closed immersion induces a closed immersion

[math]\displaystyle{ \mathbf{P}(\mathcal G_T) \to \mathbf{P} (\mathcal{E}) \times_S T. }[/math]

Conversely, any such closed immersion comes from a surjective homomorphism of [math]\displaystyle{ O_T }[/math]-modules from [math]\displaystyle{ \mathcal E_T }[/math] to a locally free module of rank [math]\displaystyle{ k }[/math]. Therefore, the elements of [math]\displaystyle{ \mathbf{Gr}(k, \mathcal E)(T) }[/math] are exactly the projective subbundles of rank [math]\displaystyle{ k }[/math] in [math]\displaystyle{ \mathbf{P} (\mathcal{E}) \times_S T. }[/math]

Under this identification, when [math]\displaystyle{ T=S }[/math] is the spectrum of a field [math]\displaystyle{ K }[/math] and [math]\displaystyle{ \mathcal E }[/math] is given by a vector space [math]\displaystyle{ V }[/math], the set of rational points [math]\displaystyle{ \mathbf{Gr}(k, \mathcal{E})(K) }[/math] correspond to the projective linear subspaces of dimension [math]\displaystyle{ k-1 }[/math] in [math]\displaystyle{ \mathbf{P}(V) }[/math], and the image of [math]\displaystyle{ \mathbf{P}(\mathcal G)(K) }[/math] in

[math]\displaystyle{ \mathbf{P}(V) \times_K \mathbf{Gr}(k, \mathcal E) }[/math]

is the set

[math]\displaystyle{ \left\{ (x, v) \in \mathbf{P}(V)(K) \times \mathbf{Gr}(k, \mathcal E)(K) \mid x\in v \right\}. }[/math]

The Plücker embedding

The Plücker embedding is a natural embedding of the Grassmannian [math]\displaystyle{ \mathbf{Gr}(k, V) }[/math] into the projectivization of the [math]\displaystyle{ k }[/math]th Exterior power [math]\displaystyle{ \Lambda^k V }[/math] of [math]\displaystyle{ V }[/math].

[math]\displaystyle{ \iota : \mathbf{Gr}(k, V) \to \mathbf{P} \left(\Lambda^k V \right ). }[/math]

Suppose that [math]\displaystyle{ w\subset V }[/math] is a [math]\displaystyle{ k }[/math]-dimensional subspace of the [math]\displaystyle{ n }[/math]-dimensional vector space [math]\displaystyle{ V }[/math]. To define [math]\displaystyle{ \iota(w) }[/math], choose a basis [math]\displaystyle{ (w_1, \cdots, w_k) }[/math] for [math]\displaystyle{ w }[/math], and let [math]\displaystyle{ \iota(w) }[/math] be the projectivization of the wedge product of these basis elements: [math]\displaystyle{ \iota(w) = [w_1 \wedge \cdots \wedge w_k], }[/math] where [math]\displaystyle{ [ \, \cdot \, ] }[/math] denotes the projective equivalence class.

A different basis for [math]\displaystyle{ w }[/math] 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, [math]\displaystyle{ \iota }[/math] is well-defined. To see that it is an embedding, notice that it is possible to recover [math]\displaystyle{ w }[/math] from [math]\displaystyle{ \iota(w) }[/math] as the span of the set of all vectors [math]\displaystyle{ v\in V }[/math] such that

[math]\displaystyle{ v \wedge \iota (w) = 0 }[/math].

Plücker coordinates and Plücker relations

The Plücker embedding of the Grassmannian satisfies a set of simple quadratic relations called the Plücker relations. These show that the Grassmannian [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] embeds as a nonsingular projective algebraic subvariety of the projectivization [math]\displaystyle{ \mathbf{P}(\Lambda^k V) }[/math] of the [math]\displaystyle{ k }[/math]th exterior power of [math]\displaystyle{ V }[/math] and give another method for constructing the Grassmannian. To state the Plücker relations, fix a basis [math]\displaystyle{ (e_1, \cdots, e_n) }[/math] for [math]\displaystyle{ V }[/math], and let [math]\displaystyle{ w\subset V }[/math] be a [math]\displaystyle{ k }[/math]-dimensional subspace of [math]\displaystyle{ V }[/math] with basis [math]\displaystyle{ (w_1, \cdots, w_k) }[/math]. Let [math]\displaystyle{ (w_{i1}, \cdots, w_{in}) }[/math] be the components of [math]\displaystyle{ w_i }[/math] with respect to the chosen basis of [math]\displaystyle{ V }[/math], and [math]\displaystyle{ (W^1, \dots, W^n) }[/math] the [math]\displaystyle{ k }[/math]-component column vectors forming the transpose of the corresponding homogeneous coordinate matrix:

[math]\displaystyle{ W^T = [W^1\, \cdots W^n]= \begin{bmatrix} w_{11} &\cdots & w_{1n}\\ \vdots & \ddots & \vdots\\ w_{k1} & \cdots & w_{kn} \end{bmatrix} , }[/math]

For any ordered sequence [math]\displaystyle{ 1\le i_1 \lt \cdots \lt i_k \le n }[/math] of [math]\displaystyle{ k }[/math] positive integers, let [math]\displaystyle{ w_{i_1, \dots , i_k} }[/math] be the determinant of the [math]\displaystyle{ k \times k }[/math] matrix with columns [math]\displaystyle{ [W^{i_1}, \dots , W^{i_k}] }[/math]. The elements [math]\displaystyle{ \{w_{i_1, \dots , i_k} \, \vert \, 1 \leq i_1 \lt \cdots \lt i_k \leq n\} }[/math] are called the Plücker coordinates of the element [math]\displaystyle{ w \in \mathbf{Gr}_k(V) }[/math] of the Grassmannian (with respect to the basis [math]\displaystyle{ (e_1, \cdots, e_n) }[/math] of [math]\displaystyle{ V }[/math]). These are the linear coordinates of the image [math]\displaystyle{ \iota(w) }[/math] of [math]\displaystyle{ w }[/math] under the Plücker map, relative to the basis of the exterior power [math]\displaystyle{ \Lambda^k V }[/math] space generated by the basis [math]\displaystyle{ (e_1, \cdots, e_n) }[/math] of [math]\displaystyle{ V }[/math]. Since a change of basis for [math]\displaystyle{ w }[/math] 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 [math]\displaystyle{ \mathbf{P}(\Lambda^k V) }[/math].

For any two ordered sequences [math]\displaystyle{ 1 \leq i_1 \lt i_2 \cdots \lt i_{k-1} \leq n }[/math] and [math]\displaystyle{ 1 \leq j_1 \lt j_2 \cdots \lt j_{k+1} \leq n }[/math] of [math]\displaystyle{ k-1 }[/math] and [math]\displaystyle{ k+1 }[/math] positive integers, respectively, the following homogeneous quadratic equations, known as the Plücker relations, or the Plücker-Grassmann relations, are valid and determine the image [math]\displaystyle{ \iota(\mathbf{Gr}_k(V)) }[/math] of [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] under the Plücker map embedding:

[math]\displaystyle{ \sum_{l=1}^{k+1} (-1)^\ell w_{i_1, \dots , i_{k-1}, j_l} w_{j_1, \dots , \widehat{j_l}, \dots j_{k+1}} = 0, }[/math]

where [math]\displaystyle{ j_1, \ldots , \widehat{j_l}, \ldots j_{k+1} }[/math] denotes the sequence [math]\displaystyle{ j_1, \ldots, j_{k+1} }[/math] with the term [math]\displaystyle{ j_l }[/math] omitted. These are consistent, determining a nonsingular projective algebraic variety, but they are not algebraically independent. They are equivalent to the statement that [math]\displaystyle{ \iota(w) }[/math] is the projectivization of a completely decomposable element of [math]\displaystyle{ \Lambda^k V }[/math].

When [math]\displaystyle{ \dim(V) =4 }[/math], and [math]\displaystyle{ k=2 }[/math] (the simplest Grassmannian that is not a projective space), the above reduces to a single equation. Denoting the homogeneous coordinates of the image [math]\displaystyle{ \iota(\mathbf{Gr}_2(V) \subset \mathbf{P}(\Lambda^2 V) }[/math] under the Plücker map as [math]\displaystyle{ (w_{12}, w_{13}, w_{14}, w_{23}, w_{24}, w_{34}) }[/math], this single Plücker relation is

[math]\displaystyle{ w_{12}w_{34} - w_{13}w_{24} + w_{14}w_{23} = 0. }[/math]

In general, many more equations are needed to define the image [math]\displaystyle{ \iota(\mathbf{Gr}_k(V)) }[/math] of the Grassmannian in [math]\displaystyle{ \mathbf{P}(\Lambda^k V) }[/math] under the Plücker embedding.

Duality

Every [math]\displaystyle{ k }[/math]-dimensional subspace [math]\displaystyle{ W \subset V }[/math] determines an [math]\displaystyle{ (n-k) }[/math]-dimensional quotient space [math]\displaystyle{ V/W }[/math] of [math]\displaystyle{ V }[/math]. This gives the natural short exact sequence:

[math]\displaystyle{ 0 \rightarrow W \rightarrow V \rightarrow V/W \rightarrow 0. }[/math]

Taking the dual to each of these three spaces and the dual linear transformations yields an inclusion of [math]\displaystyle{ (V/W)^* }[/math] in [math]\displaystyle{ V^* }[/math] with quotient [math]\displaystyle{ W^* }[/math]

[math]\displaystyle{ 0 \rightarrow (V/W)^* \rightarrow V^* \rightarrow W^* \rightarrow 0. }[/math]

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 [math]\displaystyle{ k }[/math]-dimensional subspaces of [math]\displaystyle{ V }[/math] and [math]\displaystyle{ (n-k) }[/math]-dimensional subspaces of [math]\displaystyle{ V^* }[/math]. In terms of the Grassmannian, this gives a canonical isomorphism

[math]\displaystyle{ \mathbf{Gr}_k(V) \leftrightarrow \mathbf{Gr}{(n-k}, V^*) }[/math]

that associates to each subspace [math]\displaystyle{ W \subset V }[/math] its annihilator [math]\displaystyle{ W^0\subset V^* }[/math]. Choosing an isomorphism of [math]\displaystyle{ V }[/math] with [math]\displaystyle{ V^* }[/math] therefore determines a (non-canonical) isomorphism between [math]\displaystyle{ \mathbf{Gr}_k( V) }[/math] and [math]\displaystyle{ \mathbf{Gr}_{n-k}(V) }[/math]. An isomorphism of [math]\displaystyle{ V }[/math] with [math]\displaystyle{ V^* }[/math] is equivalent to the choice of an inner product, so with respect to the chosen inner product, this isomorphism of Grassmannians sends any [math]\displaystyle{ k }[/math]-dimensional subspace into its [math]\displaystyle{ (n-k) }[/math]}-dimensional orthogonal complement.

Schubert cells

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 [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] are defined in terms of a specified complete flag of subspaces [math]\displaystyle{ V_1 \subset V_2 \subset \cdots \subset V_n=V }[/math] of dimension [math]\displaystyle{ \mathrm{dim}(V_i) = i }[/math]. For any partition

[math]\displaystyle{ \lambda =(\lambda_1, \cdots, \lambda_k) }[/math]

of weight

[math]\displaystyle{ |\lambda|=\sum_{i=1}^k\lambda_i }[/math]

consisting of weakly decreasing non-negative integers

[math]\displaystyle{ \lambda_1 \geq \cdots \geq \lambda_k \geq 0, }[/math]

whose Young diagram fits within the rectangular one [math]\displaystyle{ (n-k)^k }[/math], the Schubert cell [math]\displaystyle{ X_\lambda(k,n)\subset \mathbf{Gr}_k(V) }[/math] consists of those elements [math]\displaystyle{ W \in \mathbf{Gr}_k(V) }[/math] whose intersections with the subspaces [math]\displaystyle{ \{V_i\} }[/math] have the following dimensions

[math]\displaystyle{ X_\lambda(k,n) = \{W \in \mathbf{Gr}_k(V)\, | \, \dim(W \cap V_{n-k+j-\lambda_j}) = j\}. }[/math]

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 [math]\displaystyle{ \chi_{k,n} }[/math] of the Grassmannian [math]\displaystyle{ \mathbf{Gr}_k(\mathbf{R}^n) }[/math] of k-dimensional subspaces of Rⁿ. Fix a [math]\displaystyle{ 1 }[/math]-dimensional subspace [math]\displaystyle{ \mathbf{R}\subset \mathbf{R}^n }[/math] and consider the partition of [math]\displaystyle{ \mathbf{Gr}_k(\mathbf{R}^n) }[/math] into those k-dimensional subspaces of Rⁿ that contain R and those that do not. The former is [math]\displaystyle{ \mathbf{Gr}_{k-1}(\mathbf{R}^{n-1}) }[/math] and the latter is a rank [math]\displaystyle{ k }[/math] vector bundle over [math]\displaystyle{ \mathbf{Gr}_k(\mathbf{R}^{n-1}) }[/math]. This gives recursive formulae:

[math]\displaystyle{ \chi_{k,n} = \chi_{k-1,n-1} + (-1)^k \chi_{k, n-1}, \qquad \chi_{0,n} = \chi_{n,n} = 1. }[/math]

Solving these recursion relations gives the formula: [math]\displaystyle{ \chi_{k,n}=0 }[/math] if [math]\displaystyle{ n }[/math] is even and [math]\displaystyle{ k }[/math] is odd and

[math]\displaystyle{ \chi_{k, n} = \begin{pmatrix}\left\lfloor \frac{n}{2} \right\rfloor \\ \left\lfloor \frac{k}{2} \right\rfloor \end{pmatrix} }[/math]

otherwise.

Cohomology ring of the complex Grassmannian

Every point in the complex Grassmann manifold [math]\displaystyle{ \mathbf{Gr}_k(\mathbf{C}^n) }[/math] defines a [math]\displaystyle{ k }[/math]-plane in [math]\displaystyle{ n }[/math]-space. Fibering these planes over the Grassmannian one arrives at the vector bundle [math]\displaystyle{ E }[/math] which generalizes the tautological bundle of a projective space. Similarly the [math]\displaystyle{ (n-k) }[/math]-dimensional orthogonal complements of these planes yield an orthogonal vector bundle [math]\displaystyle{ F }[/math]. The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of [math]\displaystyle{ E }[/math]. 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 [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math]. Then the relations merely state that the direct sum of the bundles [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math] is trivial. Functoriality of the total Chern classes allows one to write this relation as

[math]\displaystyle{ c(E) c(F) = 1. }[/math]

The quantum cohomology ring was calculated by Edward Witten. The generators are identical to those of the classical cohomology ring, but the top relation is changed to

[math]\displaystyle{ c_k(E) c_{n-k}(F) = (-1)^{n-k} }[/math]

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

Associated measure

When [math]\displaystyle{ V }[/math] is an [math]\displaystyle{ n }[/math]-dimensional Euclidean space, we may define a uniform measure on [math]\displaystyle{ \mathbf{Gr}_k(V) }[/math] in the following way. Let [math]\displaystyle{ \theta_n }[/math] be the unit Haar measure on the orthogonal group [math]\displaystyle{ O(n) }[/math] and fix [math]\displaystyle{ w\in \mathbf{Gr}_k(V) }[/math]. Then for a set [math]\displaystyle{ A \subset\mathbf{Gr}_k(V) }[/math] , define

[math]\displaystyle{ \gamma_{k, n}(A) = \theta_n\{g \in \operatorname{O}(n) : gw \in A\}. }[/math]

This measure is invariant under the action of the group [math]\displaystyle{ O(n) }[/math]; that is,

[math]\displaystyle{ \gamma_{k,n}(gA)= \gamma_{k,n}(A) }[/math]

for all [math]\displaystyle{ g \in O(n) }[/math]. Since [math]\displaystyle{ \theta_n(O(n))=1 }[/math], we have [math]\displaystyle{ \gamma_{k,n}(\mathbf{Gr}_k(V))= 1 }[/math]. Moreover, [math]\displaystyle{ \gamma_{k,n} }[/math] 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 [math]\displaystyle{ k }[/math]-dimensional subspaces of [math]\displaystyle{ \mathbf{R}^n }[/math]. It is a double cover of [math]\displaystyle{ \mathbf{Gr}_k(\mathbf{R}^n) }[/math] and is denoted by [math]\displaystyle{ \widetilde{\mathbf{Gr}}_k(\mathbf{R}^n) }[/math].

As a homogeneous space it can be expressed as:

[math]\displaystyle{ \widetilde{\mathbf{Gr}}_k(\mathbf{R}^n)=\operatorname{SO}(n) / (\operatorname{SO}(k) \times \operatorname{SO}(n-k)). }[/math]

Orthogonal isotropic Grassmannians

Given a real or complex nondegenerate symmetric bilinear form [math]\displaystyle{ Q }[/math] on the [math]\displaystyle{ n }[/math]-dimensional space [math]\displaystyle{ V }[/math] (i.e., a scalar product), the totally isotropic Grassmannian [math]\displaystyle{ \mathbf{Gr}^0_k(V, Q) }[/math] is defined as the subvariety [math]\displaystyle{ \mathbf{Gr}^0_k(V, Q) \subset \mathbf{Gr}_k(V) }[/math] consisting of all [math]\displaystyle{ k }[/math]-dimensional subspaces [math]\displaystyle{ w\subset V }[/math] for which

[math]\displaystyle{ Q(u, v)=0, \, \forall \, u, v \in w. }[/math]

Maximal isotropic Grassmannians with respect to a real or complex scalar product are closely related to Cartan's theory of spinors. 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.

Applications

A key application of Grassmannians is as the "universal" embedding space for bundles with connections on compact manifolds.

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.

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. The KP equations, expressed in Hirota bilinear form in terms of the KP Tau function are equivalent to the Plücker relations. 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.

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.

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.

Grassmann manifolds have also found applications in computer vision tasks of video-based face recognition and shape recognition, and are used in the data-visualization technique known as the grand tour.

See also

• 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.
• Affine Grassmannian
• Grassmann bundle
• Grassmann graph

References

• 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. </ref>
• Hatcher, Allen (2003). Vector Bundles & K-Theory (2.0 ed.). https://www.maths.ed.ac.uk/~v1ranick/papers/hatchvb.pdf.  section 1.2
• Characteristic classes. Annals of Mathematics Studies. 76. Princeton, NJ: Princeton University Press. 1974. 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. https://books.google.com/books?id=k91UpG26Hp8C. 
• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 978-0-8218-2848-9 
• Mattila, Pertti (1995). Geometry of Sets and Measures in Euclidean Spaces. New York: Cambridge University Press. ISBN 0-521-65595-1. https://books.google.com/books?id=RTQmlDw9r6gC. 
• Shafarevich, Igor R. (2013). Basic Algebraic Geometry 1. Springer Science. doi:10.1007/978-3-642-37956-7. ISBN 978-0-387-97716-4. https://link.springer.com/book/10.1007/978-3-642-37956-7.