Wikipedia
In mathematics, the Grassmannian is a space which parameterizes all linear subspaces of a vector space of given dimension . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than .
When is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety.
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 being equivalent to , and with some authors using or to denote the Grassmannian of -dimensional subspaces of an unspecified -dimensional vector space.
In mathematics, a Grassmannian may refer to:
- Affine Grassmannian
- Affine Grassmannian (manifold)
- Grassmannian, the classical parameter space for linear subspaces of a linear space or projective space
- Lagrangian Grassmannian