What do cluster algebras tell us about Grassmannians?

Question

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.

Answer 1

I'm afraid that as far as I know, the answer is no. That is, the cluster structure hasn't (yet) told us anything new. There are two reasons why we might have expected that, though.

Firstly, the Grassmannians have been studied over a long period of time in many different settings, in which much of the time this same phenomenon has happened - Grassmannians provide the nice, small examples, that we understand "completely". So asking the cluster structure to prove something new is quite a big ask!

Secondly, cluster algebras (and their quantum analogues) haven't been around very long in mathematical terms. There are very few results of the form "Let $X$ be a cluster algebra. Then $X$ is a $Y$.", where $Y$ is a ring or algebra property.

Some results this direction include recent work of Phillip Lampe, but these are the only ones that spring to mind. Others may know of more, of course.

I have also given a number of cluster algebra talks, mostly about quantum Grassmannians, and my answer has been to point out that precisely because Grassmannians turn up everywhere, they are ideal for looking at as a first set of examples. Just as a group theorist might say "and this is what it looks like for Abelian groups", or $p$-groups, say. It's natural when studying projective varieties to turn to Grassmannians as the first non-trivial examples.

Answer 2

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.