Grassmannian - Imported Knowlpedia Content

Definition

Let $V$ be an $n$ -dimensional vector space over $\mathbb F=\mathbb R$ or $\mathbb C$ , and let $0\leq k\leq n$ . The Grassmannian

\operatorname{Gr}_k(V)=\{W\subseteq V:\dim_{\mathbb F}W=k\}

is the set of all $k$ -dimensional linear subspaces of $V$ , equipped with its natural smooth-manifold structure. It has real dimension $k(n-k)$ when $\mathbb F=\mathbb R$ , and complex dimension $k(n-k)$ when $\mathbb F=\mathbb C$ . The cases $k=0,n$ are single points.

Local charts and tangent spaces

Choose a decomposition $V=W\oplus W'$ . Every $k$ -plane sufficiently near $W$ is uniquely the graph of a linear map $A:W\to W'$ , giving a chart modeled on $\operatorname{Hom}(W,W')$ . Intrinsically, the tangent space at $W$ is

T_W\operatorname{Gr}_k(V)\cong\operatorname{Hom}(W,V/W).

These charts exhibit the stated dimension and make the construction independent of a chosen basis.

Homogeneous-space models

After choosing an inner product, the real Grassmannian is

\operatorname{Gr}_k(\mathbb R^n)\cong O(n)/(O(k)\times O(n-k)).

With a Hermitian inner product, the complex Grassmannian is $U(n)/(U(k)\times U(n-k))$ . These presentations show that the Grassmannians are compact smooth manifolds and relate their geometry to transitive Lie-group actions.

Algebraic and bundle structure

For complex $V$ , the Plücker map sends $W$ to the line $\bigwedge^kW\subseteq\bigwedge^kV$ , realizing the Grassmannian as a smooth projective algebraic variety cut out by quadratic Plücker relations Harris, Lecture 6. Over the Grassmannian, the fibers $W$ themselves form the tautological rank- $k$ vector bundle.

References

J. Harris, Algebraic Geometry: A First Course, Springer, 1992. Springer DOI record. Relevant: Lecture 6.
J. Milnor and J. Stasheff, Characteristic Classes, Princeton University Press, 1974. Princeton DOI record. Relevant: §5 and Grassmannians as classifying spaces.