Plücker embedding

In mathematics, the Plücker map embeds the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$, whose elements are k-dimensional subspaces of an n-dimensional vector space V, either real or complex, in a projective space, thereby realizing it as a projective algebraic variety. More precisely, the Plücker map embeds ${\displaystyle \mathbf {Gr} (k,V)}$ into the projectivization ${\displaystyle \mathbf {P} (\Lambda ^{k}V)}$ of the ${\displaystyle k}$-th exterior power of ${\displaystyle V}$. The image is algebraic, consisting of the intersection of a number of quadrics defined by the § Plücker relations (see below).

The Plücker embedding was first defined by Julius Plücker in the case ${\displaystyle k=2,n=4}$ as a way of describing the lines in three-dimensional space (which, as projective lines in real projective space, correspond to two-dimensional subspaces of a four-dimensional vector space). The image of that embedding is the Klein quadric in RP⁵.

Hermann Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$ under the Plücker embedding, relative to the basis in the exterior space ${\displaystyle \Lambda ^{k}V}$ corresponding to the natural basis in ${\displaystyle V=K^{n}}$ (where ${\displaystyle K}$ is the base field) are called Plücker coordinates.