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?



Isomorphism between some GIT quotients
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.


