Plücker embedding

This article is about the general Plücker embedding. For the classical case of 2-planes in 4-space, see Plücker coordinates.

In mathematics, the Plücker embedding describes 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). This was generalized by Hermann Grassmann to arbitrary r and n using a generalization of Plücker's coordinates, sometimes called Grassmann coordinates.

Definition

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

\begin{align} \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{align}

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, ..., wr}, and {z1, ..., zr}, 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 X1,2, X1,3, X1,4, X2,3, X2,4, X3,4, we have that Gr(2, V) is defined by the equation

X1,2X3,4 − X1,3X2,4 + X2,3X1,4 = 0.

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

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

Further reading