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In mathematics, the Plücker map embeds the
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
\mathbf(k,V), whose elements are ''k''- dimensional subspaces of an ''n''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
''V'', in a projective space, thereby realizing it as an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
. More precisely, the Plücker map embeds \mathbf(k,V) into the projectivization \mathbf(\Lambda^k V) of the k-th exterior power of 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 Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the di ...
in the case k=2, n= 4 as a way of describing the lines in three-dimensional space (which, as
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
s 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 mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein qu ...
in RP5.
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
generalized Plücker's embedding to arbitrary ''k'' and ''n''. The homogeneous coordinates of the image of the Grassmannian \mathbf(k,V) under the Plücker embedding, relative to the basis in the exterior space \Lambda^k V corresponding to the natural basis in V = K^n (where K is the base field) are called Plücker coordinates.


Definition

Denoting by V= K^n the n-dimensional vector space over the field K, and by \mathbf(k, V) the Grassmannian of k-dimensional subspaces of V, the Plücker embedding is the map ''ι'' defined by ::\begin \iota \colon \mathbf(k, V) &\rightarrow \mathbf(\Lambda^k V),\\ \iota \colon w:=\operatorname( w_1, \ldots, w_k ) &\mapsto w_1 \wedge \cdots \wedge w_k \end where (w_1, \dots , w_k) is a basis for the element w\in \mathbf(k, V) and w_1 \wedge \cdots \wedge w_k /math> is the projective equivalence class of the element w_1 \wedge \cdots \wedge w_k \in \Lambda^k V of the kth exterior power of V. This is an embedding of the Grassmannian into the projectivization \mathbf(\Lambda^k V). 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...
. The
bracket ring In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials ''k'' 'x''11,...,''x'dn''generated by the ''d''-by-''d'' minors of a generic ''d''-by-''n'' matrix (''x'ij''). The bracket ring may be regarded as th ...
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 \mathbf(\Lambda^k V) 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 W_1, \dots, W_k. Let W be the n \times k matrix of homogeneous coordinates, whose columns are W_1, \dots, W_k. For any ordered sequence 1\le i_1 < \cdots < i_k \le n of k integers, let W_ be the determinant of the k \times k matrix whose rows are the i_1, \dots i_k rows of W . Then, up to projectivization, \ are the Plücker coordinates of the element /math> of the Grassmannian \mathbf(k, V) whose homogeneous coordinates are W. They are the linear coordinates of the image \iota(W) of \in \mathbf(k, V) under the Plücker map, relative to the standard basis in the exterior space \Lambda^k V . For any two ordered sequences: :: i_1 < i_2 < \cdots < i_, \quad j_1 < j_2 < \cdots < j_ of positive integers 1 \le i_l, j_m \le n , the following homogeneous equations are valid, and determine the image of under the Plücker map: where j_1, \dots , \hat_l \dots j_ denotes the sequence j_1, \dots , \dots j_ with the term j_l omitted. When and , the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of \Lambda^2 V by :: W_ = -W_, \quad 1\le i,j, \le 4, the image of \mathbf(2, V) under the Plücker map is defined by the single equation :: W_W_ - W_W_ + W_W_=0. 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 In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically i ...
.


References


Further reading

* {{DEFAULTSORT:Plucker embedding Algebraic geometry Differential geometry