corresponding $n$-simplex.

The simply laced case
---------------------

This case was considered in [@Ke07]. We review the results here in order to introduce the notations.

If ${{\mathfrak{g}}}$ is simply-laced, the Cartan matrix is symmetric, $C^t=C$, and therefore the matrix $A$ vanishes and $B$ is block-diagonal: $$\label{Bsimple}
B=\left(\begin{array}{rr} 0 & -C \\ C & 0 \end{array}\right).$$

The graph ${\mathcal G}_{{\mathfrak{g}}}$ has a particularly simple form in this case. It includes the special nodes $k$ ($k\in {{\mathbb Z}}_+$) with cluster variables $$\label{simpleclus}
{{\mathbf x}}[k]
= ({{R}}_{1,2k},...,{{R}}_{r,2k},{{R}}_{1,2k+1},...,{{R}}_{r,2k+1}).$$ In addition, it includes the special nodes $k'$, which are reached from $k$ by the compound mutation $\mu^{(1)} = \mu_1\circ \cdots \circ
\mu_r$. These commute due to the block-diagonal structure of $B$. The following was proved in [@Ke07]:

\[simple\] For $k\in {{\mathbb Z}}_+$, Let ${{\mathbf x}}[k]$ be as in and $B$ be the matrix in . Then

1.  At the node $k$, the mutations $\mu_1,...,\mu_r$ commute among themselves, and evolution according to these mutations corresponds to the $Q$-system evolutions ${{R}}_{{{\alpha}},2k}\mapsto {{R}}_{{{\alpha}},2k+2}$, ${{\alpha}}\in I_r$.

2.  At the same node, the mutations $\mu_{r+1},...,\mu_{2r}$ also commute among themselves, and applying them to the variables ${{\mathbf x}}[k]$ corresponds to the $Q$-system evolution ${{R}}_{{{\alpha}},2k+1}\mapsto {{R}}_{{{\alpha}},2k-1}$, ${{\alpha}}\in I_r$.

3.  Define the compound mutation 