Grassmannian

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

When $V$ 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 $r(n-r)$ .

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 $Gr (V, r)$ being equivalent to $Gr (r, V)$ , and with some authors using $Gr (r, n)$ or $Gr (n, r)$ to denote the Grassmannian of $r$ -dimensional subspaces of an unspecified $n$ -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 $M$ of dimension $r$ embedded in $R n$ . At each point $x$ in $M$ , the tangent space to $M$ can be considered as a subspace of the tangent space of $R n$ , which is just $R n$ . The map assigning to $x$ its tangent space defines a map from $M$ to $Gr (r, n)$ . (In order to do this, we have to translate the geometrical tangent space to $M$ so that it passes through the origin rather than $x$ , and hence defines a $r$ -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 $M$ , so that every vector bundle generates a continuous map from $M$ 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 $r = 1$ , 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 of n−1 dimensions.

For $r = 2$ , the Grassmannian is the space of all planes through the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin perpendicular to that plane (and vice versa); hence $Gr (2, 3) ≅ Gr (1, 3) ≅ P 2$ , the projective plane.

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

The Grassmannian as a set

Let $V$ be a finite-dimensional vector space over a field $k$ . The Grassmannian $Gr (r, V)$ is the set of all $r$ -dimensional linear subspaces of $V$ . If $V$ has dimension $n$ , then the Grassmannian is also denoted $Gr (r, n)$ .

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 $GL(V)$ acts transitively on the $r$ -dimensional subspaces of $V$ . Therefore, if $H$ is the stabilizer of any of the subspaces under this action, we have

Gr (r, V) = GL(V)/ H

If the underlying field is $R$ or $C$ and $GL(V)$ 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 $V$ . Over $R$ , one replaces $GL(V)$ by the orthogonal group $O(V)$ , and by restricting to orthonormal frames, one gets the identity

Gr (r, n) = O(n)/(O(r) \times O(n - r))

In particular, the dimension of the Grassmannian is $r (n - r)$ .

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

d(W, W') = \lVert P_W - P_{W'} \rVert,

where $||\cdot||$ denotes the operator norm, is a metric on $Gr (r, V)$ . The exact inner product used does not matter, because a different inner product will give an equivalent norm on $V$ , and so give an equivalent metric.

If the ground field $k$ is arbitrary and $GL(V)$ 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, $H$ is a parabolic subgroup of $GL(V)$ .

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

\mathcal E_T := \mathcal E\otimes_{O_S} O_T

locally free of rank $r$ on $T$ . We denote this set by $\mathbf{Gr}(r, \mathcal{E}_T)$ .

This functor is representable by a separated $S$ -scheme $\mathbf{Gr}(r, \mathcal{E})$ . The latter is projective if ${\mathcal {E}}$ is finitely generated. When $S$ is the spectrum of a field $k$ , then the sheaf ${\mathcal {E}}$ is given by a vector space $V$ and we recover the usual Grassmannian variety of the dual space of $V$ , namely: $Gr (r, V *)$ .

By construction, the Grassmannian scheme is compatible with base changes: for any $S$ -scheme $S'$ , we have a canonical isomorphism

\mathbf{Gr}(r, \mathcal E) \times_S S' \simeq \mathbf{Gr}(r, \mathcal E_{S'})

In particular, for any point $s$ of $S$ , the canonical morphism ${s} = Spec(k (s)) \to S$ , induces an isomorphism from the fiber $\mathbf{Gr}(r, \mathcal E)_s$ to the usual Grassmannian ${Gr}(r, \mathcal E\otimes_{O_S} k(s))$ over the residue field $k (s)$ .

Universal family

Since the Grassmannian scheme represents a functor, it comes with a universal object, ${\mathcal {G}}$ , which is an object of

\mathbf{Gr} \left (r, \mathcal{E}_{\mathbf {Gr}(r, \mathcal E)} \right),

and therefore a quotient module ${\mathcal {G}}$ of $\mathcal E_{\mathbf {Gr}(r, \mathcal E)}$ , locally free of rank $r$ over $\mathbf{Gr}(r, \mathcal{E})$ . The quotient homomorphism induces a closed immersion from the projective bundle $\mathbf{P}(\mathcal G)$ :

\mathbf{P}(\mathcal G)\to \mathbf{P} \left (\mathcal E_{\mathbf{Gr}(r, \mathcal E)} \right ) = \mathbf P({\mathcal E})\times_S \mathbf{Gr}(r, \mathcal E).

For any morphism of $S$ -schemes:

T \to \mathbf{Gr}(r, \mathcal{E}),

this closed immersion induces a closed immersion

\mathbf{P}(\mathcal G_T) \to \mathbf{P} (\mathcal{E})\times_S T.

Conversely, any such closed immersion comes from a surjective homomorphism of $O T$ -modules from $\mathcal E_T$ to a locally free module of rank $r$ .^[3] Therefore, the elements of $\mathbf{Gr}(r, \mathcal E)(T)$ are exactly the projective subbundles of rank $r$ in

\mathbf{P} (\mathcal{E})\times_S T.

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

\mathbf{P}(V)\times_k \mathbf{Gr}(r, \mathcal E)

is the set

\{ (x, v)\in \mathbf{P}(V)(k)\times \mathbf{Gr}(r, \mathcal E)(k) \mid x\in v\}.

The Plücker embedding

The Plücker embedding is a natural embedding of a Grassmannian into a projective space:

\psi : \mathbf{Gr}(r, V) \to \mathbf{P} \left (\wedge^r V \right ).

Suppose that $W$ is an $r$ -dimensional subspace of $V$ . To define $ψ (W)$ , choose a basis ${w 1, ..., w r},$ of $W$ , and let $ψ (W)$ be the wedge product of these basis elements:

\psi(W)=w_1 \wedge \cdots \wedge w_r.

A different basis for $W$ 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, $ψ$ is well-defined. To see that $ψ$ is an embedding, notice that it is possible to recover $W$ from $ψ (W)$ as the set of all vectors $w$ such that $w \land ψ (W) = 0$ .

The embedding of the Grassmannian satisfies some very simple quadratic polynomials called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of $P (\land r V)$ and give another method of constructing the Grassmannian. To state the Plücker relations, choose two $r$ -dimensional subspaces $W$ and $Z$ of $V$ with bases ${w 1, ..., w r},$ and ${z 1, ..., z r},$ respectively. Then, for any integer $k \geq 0$ , the following equation is true in the homogeneous coordinate ring of $P (\land r V)$ :

\psi(W)\cdot\psi(Z) - \sum_{i_1 < \cdots < i_k} (v_1 \wedge \cdots \wedge v_{i_1 - 1} \wedge w_1 \wedge v_{i_1 + 1} \wedge \cdots \wedge v_{i_k - 1} \wedge w_k \wedge v_{i_k + 1} \wedge \cdots \wedge v_r)\cdot(v_{i_1} \wedge \cdots \wedge v_{i_k} \wedge w_{k+1} \cdots \wedge w_r) = 0.

When $dim(V) = 4$ , and $r = 2$ , the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of $P (\land r V)$ by $X 1,2, X 1,3, X 1,4, X 2,3, X 2,4, X 3,4$ , we have that $Gr (2, V)$ is defined by the equation

X 1,2 X 3,4 - X 1,3 X 2,4 + X 2,3 X 1,4 = 0.

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

The Grassmannian as a real affine algebraic variety

Let $Gr (r, R n)$ denote the Grassmannian of $r$ -dimensional subspaces of $R n$ . Let $M(n, R)$ denote the space of real $n \times n$ matrices. Consider the set of matrices $A (r, n) \subset M(n, R)$ defined by $X \in A (r, n)$ if and only if the three conditions are satisfied:

$X$ is a projection operator: $X 2 = X$ .
$X$ is symmetric: $X t = X$ .
$X$ has trace $r$ : $tr(X) = r$ .

$A (r, n)$ and $Gr (r, R n)$ are homeomorphic, with a correspondence established by sending $X \in A (r, n)$ to the column space of $X$ .

Duality

Every $r$ -dimensional subspace $W$ of $V$ determines an $(n - r)$ -dimensional quotient space $V / W$ of $V$ . This gives the natural short exact sequence:

0 \to W \to V \to V / W \to 0

Taking the dual to each of these three spaces and linear transformations yields an inclusion of $(V/W) *$ in $V *$ with quotient $W *$ :

0 \to (V / W) * \to V * \to W * \to 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 $r$ -dimensional subspaces of $V$ and $(n - r)$ -dimensional subspaces of $V *$ . In terms of the Grassmannian, this is a canonical isomorphism

Gr (r, V) ≅ Gr (n - r, V *)

Choosing an isomorphism of $V$ with $V *$ therefore determines a (non-canonical) isomorphism of $Gr (r, V)$ and $Gr (n - r, V)$ . An isomorphism of $V$ with $V *$ is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an $r$ -dimensional subspace into its $(n - r)$ -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 $Gr (r, n)$ are defined in terms of an auxiliary flag: take subspaces $V 1, V 2, ..., V r$ , with $V i \subset V i + 1$ . Then we consider the corresponding subset of $Gr (r, n)$ , consisting of the $W$ having intersection with $V i$ of dimension at least $i$ , for $i = 1, ..., r$ . 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 $r$ -dimensional subspaces of $R n$ . Fix a $1$ -dimensional subspace $R \subset R n$ and consider the partition of $Gr (r, n)$ into those $r$ -dimensional subspaces of $R n$ that contain $R$ and those that do not. The former is $Gr (r - 1, n - 1)$ and the latter is a $r$ -dimensional vector bundle over $Gr (r, n - 1)$ . This gives recursive formulas:

\chi_{r,n} = \chi_{r-1,n-1} + (-1)^r \chi_{r, n-1}, \qquad \chi_{0,n} = \chi_{n,n} = 1.

If one solves this recurrence relation, one gets the formula: $χ r, n = 0$ if and only if $n$ is even and $r$ is odd. Otherwise:

\chi_{r, n} = {\lfloor \frac{n}{2} \rfloor \choose \lfloor \frac{r}{2} \rfloor }.

Cohomology ring of the complex Grassmannian

Every point in the complex Grassmannian manifold $Gr (r, n)$ defines an $r$ -plane in $n$ -space. Fibering these planes over the Grassmannian one arrives at the vector bundle $E$ which generalizes the tautological bundle of a projective space. Similarly the $(n - r)$ -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 $E$ . 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 $E$ and $F$ . Then the relations merely state that the direct sum of the bundles $E$ and $F$ 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 in 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

c_k(E)c_{n-k}(F)=(-1)^{n-r}

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

Associated measure

When $V$ is $n$ -dimensional Euclidean space, one may define a uniform measure on $Gr (r, n)$ in the following way. Let $θ n$ be the unit Haar measure on the orthogonal group $O(n)$ and fix $V$ in $Gr (r, n)$ . Then for a set $A \subseteq Gr (r, n)$ , define

\gamma _{r,n}(A)=\theta _{n}\{g\in \operatorname {O} (n):gV\in A\}.

This measure is invariant under actions from the group $O(n)$ , that is, $γ r, n (gA) = γ r, n (A)$ for all $g$ in $O(n)$ . Since $θ n (O(n)) = 1$ , we have $γ r, n (Gr (r, n)) = 1$ . Moreover, $γ r, n$ 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 $r$ -dimensional subspaces of $R n$ . It is a double cover of $Gr (r, n)$ and is denoted by:

\tilde{\mathbf{Gr}}(r, n).

As a homogeneous space can be expressed as:

\operatorname {SO} (n)/(\operatorname {SO} (r)\times \operatorname {SO} (n-r)).

Applications

Grassmann manifolds have found application in computer vision tasks of video-based face recognition and shape recognition.^[4] 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.^[5]

Notes

↑ Milnor & Stasheff (1974), pp. 57–59.
↑ Grothendieck, Alexander (1971). Éléments de géométrie algébrique. 1 (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-05113-8. , Chapter I.9
↑ EGA, II.3.6.3.
↑ Pavan Taruga, 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)
↑ Arkani-Hamed, Nima; Trnka, Jaroslav (2013). "The Amplituhedron". arXiv:1312.2007 .
↑ Morel, Fabien; Voevodsky, Vladimir (1999). "A^1-homotopy theory of schemes" (PDF). Publications Mathématiques de l'IHÉS (90): 45–143. ISSN 1618-1913. MR 1813224. doi:10.1007/BF02698831. Retrieved 2008-09-05. , see section 4.3., pp. 137–140

References

Hatcher, Allen (2003). "Vector Bundles & K-Theory" (2.0 ed.). section 1.2
Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies. 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.
Mattila, Pertti (1995). Geometry of Sets and Measures in Euclidean Spaces. New York: Cambridge University Press. ISBN 0-521-65595-1.

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