Plücker embedding

In mathematics, the Plücker embedding is a method of realizing the Grassmannian G r k ( V ) {displaystyle Gr_{k}(V)} of all k-dimensional subspaces of an n-dimensional vector space V as a subvarietyof a projective space. More precisely, the Plücker map embeds G r k ( V ) {displaystyle Gr_{k}(V)} algebraically into the projective space of the k {displaystyle k} th exterior power of that vector space, P ( Λ k V ) {displaystyle P(Lambda ^{k}V)} . The image is the intersection of a number of quadrics defined by the Plücker relations. In mathematics, the Plücker embedding is a method of realizing the Grassmannian G r k ( V ) {displaystyle Gr_{k}(V)} of all k-dimensional subspaces of an n-dimensional vector space V as a subvarietyof a projective space. More precisely, the Plücker map embeds G r k ( V ) {displaystyle Gr_{k}(V)} algebraically into the projective space of the k {displaystyle k} th exterior power of that vector space, P ( Λ k V ) {displaystyle P(Lambda ^{k}V)} . The image is the intersection of a number of quadrics defined by the Plücker relations. The Plücker embedding was first defined in the case k = 2, n = 4 by Julius Plücker 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 imageof the Grassmannian G r k ( V ) {displaystyle Gr_{k}(V)} under the Plücker embedding, relative to the natural 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}} (where K {displaystyle K} is the base field) are called Plücker coordinates. The Plücker embedding (over the field K) is the map ι defined by where Gr(k, Kn) is the Grassmannian, i.e., the space of all k-dimensional subspaces of the n-dimensional vector space, Kn. This is an isomorphism from the Grassmannian to the image of ι, which is a projective variety. This variety can be completely characterized as an intersection of quadrics, each coming from a relation on the Plücker (or Grassmann) coordinates that derives from linear algebra. The bracket ring appears as the ring of polynomial functions on the exterior power. The embedding of the Grassmannian satisfies some very simple quadratic relations called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of P(∧rV) and give another method of constructing the Grassmannian. To state the Plücker relations, let W be the r-dimensional subspace spanned by the basis of row vectors {w1, ..., wr}. Let W {displaystyle W} be the r × n {displaystyle r imes n} matrix ofhomogeneous coordinates whose rows are {w1, ..., wr} and let {W1, ..., Wn}, be the corresponding column vectors. For any ordered sequence 1 ≤ i 1 < ⋯ < i k ≤ n {displaystyle 1leq i_{1}<cdots <i_{k}leq n} of k {displaystyle k} positive integers, let W i 1 , … , i k {displaystyle W_{i_{1},dots ,i_{k}}} be the determinantof the k × k {displaystyle k imes k} matrix with columns ( W i 1 , … , W i k ) {displaystyle (W_{i_{1}},dots ,W_{i_{k}})} . Then { W i 1 , … , i k } {displaystyle {W_{i_{1},dots ,i_{k}}}} are the Plücker coordinates of the element W {displaystyle W} of the Grassmannian. They are the linear coordinatesof the image ι ( W ) {displaystyle iota (W)} of W {displaystyle W} under the Plücker map, relative to the standard basis in the exterior space Λ k V {displaystyle Lambda ^{k}V}