Grassmannian

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In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr1(V) is the space of lines through the origin in V, so it is the same as the projective space PV. Grassmannians are named in honor of Hermann Grassmann.

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[edit] Motivation

By giving subspaces 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. Though such concepts may seem strangely out of place they can coincide with things that one is interested in, and can describe ideas that could not be considered otherwise—or at least describe them more economically.

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 \mathbb{R}^n. At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of \mathbb{R}^n, which is just \mathbb{R}^n. The map assigning to x its tangent space defines a map from M to Grr(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 with maps that are homotopic are isomorphic. But the definition of homotopic relies on a notion of continuity, and hence a topology.

[edit] History

The simplest Grassmannian that is not a projective space is Gr(2,4). This was studied by Julius Plücker, as lines in projective 3-space, and he parametrized the space via Plücker coordinates. Hermann Grassmann generalized Plücker's work to general r-planes in n-space.

[edit] Low dimensions

When k = 2, the Grassmannian is the space of all planes through the origin. In Euclidean 3-space, a plane is completely characterized by the one and only line perpendicular to it (and vice-versa); hence Gr2(3) is isomorphic to Gr1(3) (both of which are isomorphic to the real projective plane).

[edit] The Grassmannian as a set

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

Vector subspaces of V are equivalent to linear subspaces of the projective space PV, so it is equivalent to think of the Grassmannian as the set of all linear subspaces of PV. When the Grassmannian is thought of this way, it is often written as Grr−1(PV), Gr−1(PV), Gr(r−1, n−1), or G(r−1, n−1).

[edit] The Grassmannian as a homogenous space

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogenous 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 this action, we have

Grr(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(k) × O(n−k)).

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 PW be the projection of V onto W. Then

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

where \lVert\cdot\rVert denotes the operator norm, is a metric on Grr(V).

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 can be shown that H is a parabolic subgroup, from which it follows that Grr(V) is complete.

[edit] The Plücker embedding

Main article: Plücker embedding

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

\psi : \mbox{Gr}_r(V) \rightarrow \mathbf{P}(\wedge^r V).

Suppose that W is an r-dimensional subspace of V. To define ψ(W), choose a basis w1, ..., wr of W, and let ψ(W) be the wedge product of these basis elements:

ψ(W) = w1 ∧ ... ∧ wr.

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 ∧ 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(∧rV) 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 w1, ..., wr and z1, ..., zr, respectively. Then, for any integer k ≥ 0, the following equation is true in the homogenous coordinate ring of P(∧rV):

(w_1 \wedge \cdots \wedge w_r)\cdot(z_1 \wedge \cdots \wedge z_r) - \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.

[edit] Duality

Every r-dimensional subspace W of V determines an n−r-dimensional quotient space V/W of V. This can be written down quickly as a 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

\mbox{Gr}_r(V) \cong \mbox{Gr}_{n-r}(V^*).

Choosing an isomorphism of V with V* therefore determines a (non-canonical) isomorphism of Grr(V) and Grn−r(V). This isomorphism sends an r-dimensional subspace into its n−r-dimensional orthogonal complement.

[edit] 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 Grr(n) are defined in terms of an auxiliary flag: take subspaces V1, V2, ..., Vr, with Vi contained in Vi+1. Then we consider the corresponding subset of Grr(n), consisting of the W having intersection with Vi of dimension at least i, for i = 1 to r. The manipulation of Schubert cells is Schubert calculus.

Here is an example of the technique. Consider the problem of determining the Euler characteristic χ(Gn,r) where Gn,r is the Grassmannian of r-dimensional subspaces of \mathbb R^n. Fix a one-dimensional subspace R of \mathbb R^n and consider the partition of Gn,r into those r-dimensional subspaces of \mathbb R^n that contain R and those that do not. The former is Gn − 1,r − 1 and the latter is a r-dimensional vector bundle over Gn − 1,r. This gives recursive formulas:

\chi G_{n,r} = \chi G_{n-1,r-1} + (-1)^r \chi G_{n-1,r}\,

where by design χGn,0 = χGn,n = 1. If one solves this recurrence relation, you have the formula: χGn,r = 0 if and only if n even and r odd. Otherwise, \chi G_{n,r} = {\lfloor \frac{n}{2} \rfloor \choose \lfloor \frac{r}{2} \rfloor } .

[edit] Associated measure

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

\gamma_{n,r}(A)=\theta_{n}\{g\in O(n):gV\in A\}.

This measure is invariant under actions from the group O(n), that is, γn,r(gA) = γn,r(A) for all g\in O(n). Since θn(O(n)) = 1, we have γn,r(Gn,r) = 1. Moreover, γn,r 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.

[edit] See also

For an example of use of Grassmannians in differential geometry, see Gauss map and in projective geometry, see Plücker co-ordinates.

Flag manifolds are generalizations of Grassmannians and Stiefel manifolds are closely related.

Given a distinguished class of subspaces, one can define Grassmannians of these subspaces, such as the Lagrangian Grassmannian.

Grassmannians provide classifying spaces in K-theory, notably the classifying space for U(n)

[edit] References

  • Joe Harris, Algebraic Geometry, A First Course, (1992) Springer, New York, ISBN 0-387-97716-3
  • Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces, (1995) Cambridge University Press, New York, ISBN 0-521-65595-1
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