1 Answers
In mathematics, the Plücker map embeds the Grassmannian G r {\displaystyle \mathbf {Gr} } , whose elements are k-dimensional subspaces of an n-dimensional vector space V, in a projective space, thereby realizing it as an algebraic variety. More precisely, the Plücker map embeds G r {\displaystyle \mathbf {Gr} } into the projectivization P {\displaystyle \mathbf {P} } of the k {\displaystyle k} -th exterior power of V {\displaystyle V}. The image is algebraic, consisting of the intersection of a number of quadrics defined by the Plücker relations.
The Plücker embedding was first defined by Julius Plücker in the case k = 2 , n = 4 {\displaystyle k=2,n=4} as a way of describing the lines in three-dimensional space. The image of that embedding is the Klein quadric in RP.
Hermann Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian G r {\displaystyle \mathbf {Gr} } under the Plücker embedding, relative to the basis in the exterior space Λ k V {\displaystyle \Lambda ^{k}V} corresponding to the natural basis in V = K n {\displaystyle V=K^{n}} are called Plücker coordinates.