Plücker embedding

In mathematics, the Plücker embedding is a method to realize the Grassmannian of all r-dimensional subspaces of an n-dimensional vector space V as a subvariety of the projective space of the rth exterior power of that vector space, P(∧^r V).

The Plücker embedding was first defined, in the case r = 2, n = 4, in coordinates 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). That embedding is to the Klein quadric in RP⁵

Hermann Grassmann generalized Plucker's embedding to arbitrary r and n; the coordinate generalization is sometimes called Grassmann coordinates.

Definition

The Plücker embedding (over the field K) is the map ι defined by

${\begin{aligned}\iota \colon {\mathbf {Gr}}(r,K^{n})&{}\rightarrow {\mathbf {P}}(\wedge ^{r}K^{n})\\\operatorname {span}(v_{1},\ldots ,v_{r})&{}\mapsto K(v_{1}\wedge \cdots \wedge v_{r})\end{aligned}}$

where Gr(r, Kⁿ) is the Grassmannian, i.e., the space of all r-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 polynomials 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, choose two r-dimensional subspaces W and Z of V with bases {w₁, ..., w_r}, and {z₁, ..., z_r}, respectively. Then, for any integer k ≥ 0, the following equation is true in the homogeneous coordinate ring of P(∧^rV):

$\psi(W)\cdot\psi(Z) - \sum_{i_1 < \cdots < i_k} (v_1 \wedge \cdots \wedge v_{i_1 - 1} \wedge w_1 \wedge v_{i_1 + 1} \wedge \cdots \wedge v_{i_k - 1} \wedge w_k \wedge v_{i_k + 1} \wedge \cdots \wedge v_r)\cdot(v_{i_1} \wedge \cdots \wedge v_{i_k} \wedge w_{k+1} \cdots \wedge w_r) = 0.$

When dim(V) = 4, and r = 2, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of P(∧^rV) by X_1,2, X_1,3, X_1,4, X_2,3, X_2,4, X_3,4, we have that Gr(2, V) is defined by the equation

X_1,2X_3,4 − X_1,3X_2,4 + X_2,3X_1,4 = 0.

In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.^[2]

References

1. Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999), Oriented matroids, Encyclopedia of Mathematics and Its Applications, 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl 0944.52006
2. Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library (2nd ed.), New York: John Wiley & Sons, p. 211, ISBN 0-471-05059-8, MR 1288523, Zbl 0836.14001

Further reading

• Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. 227. New York, NY: Springer-Verlag. ISBN 0-387-23707-0. Zbl 1090.13001.