nLab
Grassmannian

Contents

Contents

Idea

For VV a vector space and rr a cardinal number (generally taken to be a natural number), the Grassmannian Gr(r,V)Gr(r,V) is the space of all rr-dimensional linear subspaces of VV.

Definition

For nn \in \mathbb{N}, write O(n)O(n) for the orthogonal group acting on n\mathbb{R}^n. More generally, say that a Euclidean vector space VV is an inner product space such that there exists a linear isometry V nV \overset{\simeq}{\to} \mathbb{R}^n to n\mathbb{R}^n equipped with its canonical inner product. Then write O(V)O(V) for the group of linear isometric automorphisms of VV. For the following we regard these groups as topological groups in the canonical way.

Definition

For n,kn, k \in \mathbb{N} and nkn \leq k, then the nnth Grassmannian of k\mathbb{R}^k is the coset topological space.

Gr n( k)O(k)/(O(n)×O(kn)), Gr_n(\mathbb{R}^k) \coloneqq O(k)/(O(n) \times O(k-n)) \,,

where the action of the product group is via its canonical embedding O(n)×O(kn)O(n)O(n)\times O(k-n) \hookrightarrow O(n).

Generally, for WVW \subset V an inclusion of Euclidean vector spaces, then

Gr W(V)O(V)/(O(W)×O(VW)), Gr_W(V) \coloneqq O(V)/(O(W)\times O(V-W)) \,,

where VWV-W denotes the orthogonal complement of WW in VV.

Remark

The group O(k)O(k) acts transitively on the set of nn-dimensional linear subspaces, and given any such, then its stabilizer subgroup in O(k)O(k) is isomorphic to O(n)×O(kn)O(n)\times O(k-n). In this way the underlying set of Gr n(k)Gr_n(k) is in natural bijection to the set of nn-dimensional linear subspaces in k\mathbb{R}^k. The realization as a coset as above serves to euqip this set naturally with a topological space.

Properties

Relation to Stiefel manifolds and universal bundles

Similarly, the real Stiefel manifold is the coset

V n(k)O(k)/O(kn). V_n(k) \coloneqq O(k)/O(k-n) \,.

The quotient projection

V n(k)Gr n(k) V_{n}(k)\longrightarrow Gr_n(k)

is an O(n)O(n)-principal bundle, with associated bundle V n(k)× O(n) nV_n(k)\times_{O(n)} \mathbb{R}^n a vector bundle of rank nn. In the limit (colimit) that kk \to \infty is this gives a presentation of the O(n)O(n)-universal principal bundle and of the universal vector bundle of rank nn, respectively.. The base space Gr n() wheBO(n)Gr_n(\infty)\simeq_{whe} B O(n) is the classifying space for O(n)O(n)-principal bundles and rank nn vector bundles.

CW-complex structure

Proposition

The real Grassmannians Gr n( k)Gr_n(\mathbb{R}^k) and the complex Grassmannians Gr n( k)Gr_n(\mathbb{C}^k) admit the structure of CW-complexes. Moreover the canonical inclusions

Gr n( k)Gr n( k+1) Gr_n(\mathbb{R}^k) \hookrightarrow Gr_n(\mathbb{R}^{k+1})

are subcomplex incusion (hence relative cell complex inclusions).

Accordingly there is an induced CW-complex structure on the classifying space

BO(n)lim kGr n( k). B O(n) \simeq \underset{\longrightarrow}{\lim}_k Gr_n(\mathbb{R}^k) \,.

A proof is spelled out in (Hatcher, section 1.2 (pages 31-34)).

Examples

Example

The Grassmannian Gr r( n+1)Gr_r(\mathbb{R}^{n+1}) (def. ) is

  • for r=0r = 0: the point;

  • for r=1r = 1: the real projective space P n\mathbb{R}P^n;

  • for r=k+1r = k+1: the point;

  • for r>k+1r \gt k+1: the empty space \emptyset.

If VV is an inner product space, then the orthogonal complement defines an isomorphism between Gr(r,V)Gr(r,V) and Gr(dimVr,V)Gr(\dim V - r,V).

References

Textbook accounts include

Lecture notes include

category: geometry, algebra

Last revised on February 8, 2017 at 13:08:58. See the history of this page for a list of all contributions to it.