Isomorphism between some GIT quotients

Question

I have a very specific question about two series of GIT quotients. Let $n\geq 2$ be an integer and let $k$ be another positive integer which is less than $n$.

Then the first series of quotients is this: consider the Grassmannian $G(k,n)$ of $k$-planes in $\mathbb C^n$. It has an action of an $n-1$-dimensional torus $T$ (maximal torus in $PGL(n)$, which is generically free. I would like to look at various GIT quotients $G(k,n)/T$. Such a quotient depends on a choice of an ample $T$-equivariant bundle; roughly speaking this is determined by $n$ integral parameters (dimension of $T$ + rank of $Pic(G(k,n))$).

On the other hand, let $X(k,n)=(\mathbb P^{k-1})^n$. It has a natural action of $PGL(k)$, which is generically free. I would like to consider all possible quotients $X(k,n)/PGL(k)$. Such a quotient also depends roughly on $n$ parameters coming from $Pic(X(k,n))$.

My question is now this: is it obvious that the two series of GIT quotients are the same?

Answer 1

Yes, it's obvious (and due to Gel$'$fand-MacPherson). Start with $k\times n$ matrices and act with $GL(k) \times T^n$, then reduce in stages in either order.

Half my thesis was about the fact that the Gel$'$fand-Cetlin system on the Grassmannian can be carried over to an integrable system on the space $X(k,n)//PGL(k)$, which for $k=2$ is a polygon space, whose integrable system was discovered independently.

Answer 2

The paper "Chow quotients of Grassmannian I" by Kapranov addresses this in some detail for the Chow quotients. See section 2.2. The relation to GIT quotients in addressed in section 0.4.