Grassmannian

In mathematics, the Grassmannian ${\displaystyle \mathbf {Gr} _{k}(V)}$ (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all ${\displaystyle k}$-dimensional linear subspaces of an ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$ over a field ${\displaystyle K}$. For example, the Grassmannian ${\displaystyle \mathbf {Gr} _{1}(V)}$ is the space of lines through the origin in ${\displaystyle V}$, so it is the same as the projective space ${\displaystyle \mathbf {P} (V)}$ of one dimension lower than ${\displaystyle V}$.^[1][2] When ${\displaystyle V}$ is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension ${\displaystyle k(n-k)}$.^[3] In general they have the structure of a nonsingular projective algebraic variety.

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to ${\displaystyle \mathbf {Gr} _{2}(\mathbf {R} ^{4})}$, parameterizing them by what are now called Plücker coordinates. (See § Plücker coordinates and Plücker relations below.) Hermann Grassmann later introduced the concept in general.

Notations for Grassmannians vary between authors, and include ${\displaystyle \mathbf {Gr} _{k}(V)}$, ${\displaystyle \mathbf {Gr} (k,V)}$,${\displaystyle \mathbf {Gr} _{k}(n)}$, ${\displaystyle \mathbf {Gr} (k,n)}$ to denote the Grassmannian of ${\displaystyle k}$-dimensional subspaces of an ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$.