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 ...
, 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 into the projectivization
of the
-th
exterior power of
. 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
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 RP
5.
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
under the Plücker embedding, relative to the basis in the exterior space
corresponding to the natural basis in
(where
is the base
field) are called
Plücker coordinates.
Definition
Denoting by
the
-dimensional vector space over the field
, and by
the Grassmannian of
-dimensional subspaces of
, the Plücker embedding is the map ''ι'' defined by
::
where
is a basis for the element
and