Grassmannian
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In mathematics, a Grassmannian \mathbf{Gr}_k(V), also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K that has a differentiable structure. For example, the Grassmannian \mathbf{Gr}_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf{P}(V) of one dimension lower than V. When V is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsin
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~31 min read
Article
25 sectionsContents
- History
- Motivation
- Low dimensions
- Differentiable manifold
- Orthogonal projections
- Affine algebraic varieties
- Homogeneous space
- Scheme
- Representable functor
- Universal family
- Plücker embedding
- Plücker coordinates and Plücker relations
- Duality
- Schubert cells
- Cohomology ring
- Associated measure
- Oriented Grassmannian
- Orthogonal isotropic Grassmannians
- Applications
- Pure mathematics
- Physics
- Computer science
- See also
- References
- Sources
In mathematics, a Grassmannian \mathbf{Gr}_k(V), also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V over a field K that has a differentiable structure. For example, the Grassmannian \mathbf{Gr}_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf{P}(V) of one dimension lower than V. When V is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The Grassmannian is named for the German polymath, linguist and mathematician Hermann Grassmann, who introduced the concept to mathematics.
== History == The earliest work on a non-trivial Grassmannian was by Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf{Gr}_2(\mathbf{R}^4), parameterizing them by what are now called Plücker coordinates. Hermann Grassmann later generalized the concept.