Grassmannian

Plücker embedding

The Plücker embedding embeds the Grassmannian of -dimensional subspaces of into the projectivization of the -th exterior power of . #m/def/geo

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

In the simplest non-projective case of we get the Klein correspondence.

Properties

• The -blades of 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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