# Understanding Grassmannian as $\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \ \text{Mat}^*_{\mathbb{R}}(k,n)$.

The Grassmannian $$\text{Gr}(k,n)$$ can be described as the quotient $$\text{Gr}(k,n) \cong \text{GL}(k,k) \ \backslash \ \text{Mat}^*_{\mathbb{R}}(k,n)$$

where $$\text{Mat}^*_{\mathbb{R}}(k,n)$$ is the set of real $$k \times n$$ matrices of rank $$k$$. Every point of $$\text{Gr}(k,n)$$ can be represented by a $$k \times n$$-matrix $$M$$. The Plücker coordinates on $$\text{Gr}(k,n)$$ are all the $$k \times k$$-minors of $$M$$.

I don't really understand this construction. I have been reading about how to construst $$\text{Gr}(k,n)$$ as a projective variety via embedding into $$\mathbb{P}\mathbb{R}$$ of dimension $$n \choose k$$ and expressing the $$n \choose k$$-minors by vanishing of homogeneous polynomials, but I am not sure how to relate that description to the one above.

I also came across a construction of $$\text{Gr}(k,n)$$ as quotient of $$\text{GL}(k,k)$$ by some stabilizer, but this seems to be related to its structure as a manifold, not as a variety, as far as I understood. My goal here is to understand another space, which is a subvariety in $$\text{Gr}(k,n)$$, but I don't really follow this definition of $$\text{Gr}(k,n)$$ to begin with.

Any help/explanation would be really great!

That should be $$\text{Mat}^*_{\mathbb{R}}(k,n)/Gl(k,k)$$. You might find it helpful to instead think of this as the quotient $$\text{Mat}^*_{\mathbb{R}}(k,n)/\sim$$, where $$\sim$$ is the relation by which $$A \sim B$$ if $$A,B$$ have the same row-space.
• Hmmm, weird that there's a typo, because it's mentioned as $\text{Gr}(k,n) \cong \text{GL}(k,k) \ / \ \text{Mat}^*_{\mathbb{R}}(k,n)$ twice in the paper I am reading..Thanks for your answer either way! – mike Jun 30 at 20:14
• @mike Is it instead written as $GL(k,k) \setminus \text{Mat}_{\Bbb R}^*(k,n)$? – Ben Grossmann Jun 30 at 20:17
• @mike The fact that it's from the left is to emphasize that the action is multiplication from the left by elements of $GL(k,k)$ – Ben Grossmann Jun 30 at 20:23
• Hmm Yes, so it's really just re-stating the same construction like with Plücker embedding? I mean, we have a map from $(k,n)$-matrices of rank $k$ to projective space of dimension $n \choose k$, where we consider the projective space in order to remain invariant under action of $\text{GL}(k,k)$, correct? – mike Jun 30 at 20:28