Plücker embedding

The Plücker embedding $𝜄 : G r 𝑘 (𝑉) ↪ P (⋀ 𝑘 𝑉)$ embeds the Grassmannian $G r 𝑘 (𝑉)$ of $𝑘$ -dimensional subspaces of $𝑉$ into the projectivization $P (⋀ 𝑘 𝑉)$ of the $𝑘$ -th exterior power of $𝑉$ . #m/def/geo

𝜄 : s p a n {𝑤 𝑖} 𝑘 𝑖 = 1 \mapsto [𝑘 ⋀ 𝑖 = 1 𝑤 𝑖]

Since the wedge product of linearly independent vectors are proportional iff they span the same linear subspace, each $𝑘$ -dimensional subspace of $𝑉$ is uniquely defined by a $𝑘$ -blade up to scaling, This motivates the embedding as given above.

In the simplest non-projective case of $G r 2 (𝕂 4)$ we get the Klein correspondence.

Properties

The $𝑘$ -blades of $𝑉$ satisfy a set of homogenous quadratic relations known as the Plücker relations. Hence the Grassmannian is embedded as a Projective variety.

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