Plücker embedding
In mathematics, the Plücker map embeds the Grassmannian formula_1, 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 formula_1 into the projectivization formula_3 of the formula_4-th exterior power of formula_5. 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 formula_6 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 RP. Hermann Grassmann generalized Plücker's embedding to arbitrary "k" and "n". The homogeneous coordinates of the image of the Grassmannian formula_1 under the Plücker embedding, relative to the basis in the exterior space formula_8 corresponding to the natural basis in formula_9 (where formula_10 is the base field) are called Plücker coordinates. Definition. Denoting by formula_11 the formula_12-dimensional vector space over the field formula_13, and by formula_14 the Grassmannian of formula_4-dimensional subspaces of  formula_16, the Plücker embedding is the map "ι" defined by where formula_18 is a basis for the element formula_19 and formula_20 is the projective equivalence class of the element formula_21 of the formula_4th exterior power of formula_5. This is an embedding of the Grassmannian into the projectivization formula_3. The image can be completely characterized as the intersection of a number of quadrics, the Plücker quadrics (see below), which are expressed by homogeneous quadratic relations on the Plücker coordinates (see below) that derive from linear algebra. The bracket ring appears as the ring of polynomial functions on the exterior power. Plücker  relations. The embedding of the Grassmannian satisfies some very simple quadratic relations usually called the Plücker relations, or Grassmann–Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of formula_25 and give another method of constructing the Grassmannian. To state the Grassmann–Plücker relations, let be the -dimensional subspace spanned by the basis of column vectors formula_26. Let formula_27 be the formula_28 matrix of homogeneous coordinates, whose columns are formula_26. For any ordered sequence formula_30 of formula_31 integers, let formula_32 be the determinant of the formula_33 matrix whose rows are the formula_34 rows of formula_27. Then, up  to projectivization, formula_36 are the Plücker coordinates of the element formula_37 of the Grassmannian formula_38 whose homogeneous coordinates are formula_39. They are the linear coordinates of the image formula_40 of formula_41 under the Plücker map, relative to the standard basis in the exterior space formula_42. For any two ordered sequences: of positive integers formula_44, the following homogeneous equations are valid, and determine the image of under the Plücker map: =0, where formula_45 denotes the sequence formula_46 with the term formula_47 omitted. When and , the simplest Grassmannian which is not a projective space, the above reduces to a single equation . Denoting the coordinates of formula_48 by the image of formula_50 under the Plücker map is defined by the single equation In general, many more equations, as in (), are needed to define the image of the Plücker embedding although these are not, in general, algebraically independent.