TITLE: Grassmannian cluster algebra of infinite type has no trees in its mutation class
QUESTION [8 upvotes]: The question is why the statement in the title is true (is it?).
To elaborate, recall that Grassmannian cluster algebra, according to Scott`s paper Grassmannians and Cluster Algebras, is the cluster structure on $\mathbb{C}[\operatorname{Gr}(k,n)]$, corresponding to the following quiver, called $\Gamma_{k,n}$:

In case $k=2$ this is a chain of length $n-3$, so the corresponding cluster algebra is of type $\mathsf{A}_{n-3}$. It is also not hard to show that for $k=3$ and $n=6,7,8$ a suitable sequence of mutations transforms $\Gamma_{k,n}$ into a tree, which happens to be a Dynkin diagram of type $\mathsf{D}_4$, $\mathsf{E}_6$ or $\mathsf{E}_8$, respectively. In all other cases this cluster algebras is of infinite type.
Now I ran some computer experiments and it seems that in the infinite case the mutation class of $\Gamma_{k,n}$ contains no trees at all.

Q1: Does this experimental observation hold in all cases? If so, why?


Q2: Can one at least find some nice representative in each mutation class? Say, having the smallest number of (unoriented) cycles.

REPLY [3 votes]: Regarding question Q2, one can go a little bit further and describe simple diagrams with few edges for some more cases.
Let us talk about $Gr(p, p+q)$, so that there is a symmetry between $p$ and $q$. Assume moreover that $p\geq 2$ and $q\geq 2$.
For $p=2$ and $2\leq q$, one can mutate to the tree $A_{q-1}$.
For $p=3$ and $2\leq q\leq 7$, one can mutate to $A_2, D_4$, $E_6$, $E_8$, $E^{(1,1)}_8$  and $K_{12}$.
for $p=4$ and $2\leq q\leq 5$, one can mutate to $A_3, E_6$, $E^{(1,1)}_7$ and $W_{12}$.
The symbols $E^{(1,1)}_8$ and $E^{(1,1)}_7$ stand for quivers related to elliptic singularities. These quivers are obtained from a tree by replacing the unique triple point by a double edge surrounded by three oriented triangles.
The symbols $K_{12}$ and $W_{12}$ stands for quivers related to two of the 14 unimodal singularities of Arnold. Unoriented graphs describing these singularities can be found in many references about singularity theory. For a picture, see page 3 of arXiv:0708.0210. The quivers are almost trees, and only have one double edge and three oriented triangles.