TITLE: What do cluster algebras tell us about Grassmannians?
QUESTION [33 upvotes]: One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. It has also been shown by Scott that $\mathbb{C}[G_{k,n}]$ carries a cluster algebra structure for any $k$.
When I'm explaining cluster algebras to somebody for the first time, I usually start with these examples, because they're the ones I spend the most time thinking about. Another reason is that the $\mathbb{C}[G_{2,n}]$ case can be nicely visualized by identifying clusters of Plücker coordinates with triangulations of the $n$-gon.
A common response to this example is "that's very pretty, but what does it tell us about $\mathbb{C}[G_{2,n}]$ as an algebra?", which is a reasonable question I don't feel I can answer well, if at all. As I'm probably going to be giving several talks about cluster algebras in the next few months, it would be nice to have a good answer to this.
Essentially all I can say so far is that the cluster monomials form a distinguished linearly independent set, and in the case of $\mathbb{C}[G_{2,n}]$ they are even a basis, but this isn't hugely satisfactory.
So, my question is:

Are there any results about
  $\mathbb{C}[G_{k,n}]$ proved using the
  cluster algebra structure that weren't
  known before this additional structure
  was discovered?

If there are better answers for different algebras then I would also be interested, but for the purposes of asking a hopefully-not-too-vague question, I'll stick to Grassmannians.

REPLY [17 votes]: One simple answer is to talk about the totally positive part of $(G_{k,n})_{> 0}$, the part of the Grassmannian where all the maximal minors (=Plücker coordinates) are real and positive.  Naively, if you want to test whether a point is in the totally positive part, you would check all the Plücker coordinates.  Since there is a cluster structure which includes all the Plücker coordinates as cluster variables, though, it suffices to check positivity with respect to any single cluster of them.
This might be a bit more geometrical than you were looking for, but I think it's a nice, and easy-to-explain, idea, which is also consistent with the roots of the development of cluster algebras.