- Definition
- Plücker relations
- References
- Further reading
In mathematics, the Plücker embedding is a method of realizing the Grassmannian 
of all k-dimensional subspaces of an n-dimensional vector space V as a subvariety
of a projective space. More precisely, the Plücker map embeds algebraically into the projective space of the 
th exterior power of that vector space, 
. 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 image
of the Grassmannian under the Plücker embedding, relative to the natural basis in the exterior space
corresponding to the natural basis in 
(where 
is the base field) are called Plücker coordinates.
Definition
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.[1]
Plücker relations
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 {{math|P(∧rV)}} and give another method of constructing the Grassmannian. To state the Plücker relations, let {{mvar|W}} be the {{mvar|r}}-dimensional subspace spanned by the basis of row vectors {{math|{w1, ..., wr}}}. Let 
be the 
matrix of
homogeneous coordinates whose rows are {{math|{w1, ..., wr}}} and let {{math|{W1, ..., Wn},}} be the corresponding column vectors. For any ordered sequence 
of positive integers, let 
be the determinant
of the 
matrix with columns 
. Then 
are the Plücker coordinates of the element of the Grassmannian. They are the linear coordinates
of the image 
of under the Plücker map, relative to the standard basis in the exterior space
For any two ordered sequences:
of positive 
integers 
, the following homogeneous equations are valid, and determine the image of {{mvar|W}} under the Plücker map:
where 
denotes
the sequence 
with
the term 
omitted.
When {{math|dim(V) {{=}} 4}}, and {{math|r {{=}} 2}}, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of {{math|P(∧rV)}} by {{math|W12, W13, W14, W23, W24, W34}}, the image of {{math|Gr(2, V)}} under the Plücker map is defined by the single equation
{{math|W12W34 − W13W24 + W23W14 {{=}} 0.}}
In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.[2]
References
2. ^{{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | page=211| publisher=John Wiley & Sons | location=New York | series=Wiley Classics Library | edition=2nd | isbn=0-471-05059-8 | mr=1288523 | year=1994 | zbl=0836.14001 }}
Further reading
- {{ cite book | last1=Miller | first1=Ezra | last2=Sturmfels | first2=Bernd | author2-link=Bernd Sturmfels | title=Combinatorial commutative algebra | series=Graduate Texts in Mathematics | volume=227 | location=New York, NY | publisher=Springer-Verlag | isbn=0-387-23707-0 | year=2005 | zbl=1090.13001 }}