Representation of elements of the Grassmannian space

[mnb96]

Hi,

I am studying some material related to Grassmannians and in particular how to represent k-subspaces of ℝⁿ as "points" in another space.

I think understood the general idea behind the Plücker embedding, however, I recently came across another type of embedding (the "Projection embedding") that sounds more intuitive and simpler to understand (see attached figure for its definition).

Can anyone elaborate a bit more on the main differences between Plücker and Projection embeddings?

In the past I browsed some old textbooks in the classical literature of algebraic geometry, and while the Plücker embedding is always treated extensively, the Projection embedding is not even mentioned at all. Why?

 

[mathwonk]

you need to define some of the symbols in your post to make it possible for us to comment. e.g what is X ? if a matrix, how does it represent an element of the grassmannian? also the text you quote contradicts your statement that the projection method is less studied than the plucker one. why not just look at some of the references in [8], which you do not give us.