\epsilon_\beta^{C_{{{\alpha}},\beta}} = -1$$ then is satisfied. But since $C$ is a Cartan matrix of finite type, it is invertible, and this system has a solution. Namely if we set $\epsilon_\beta = e^{i
  \pi \mu_\beta}$, then $\mu_{{\alpha}}= \sum_\beta C^{-1}_{\beta,{{\alpha}}}$.

In fact, we find $\epsilon_\beta^4=1$ in general. We note that this choice of normalization is done in order to make contact with cluster algebras without coefficients below. We do not believe that it is essential for the final result.

We will refer to as the normalized $Q$-system or simply the $Q$-system when it is clear that we are considering the evolution of the variables ${{R}}$.

Cluster algebras
----------------

We use here the definition of [@FZfour] to define a cluster algebra without coefficients.

[A cluster algebra.]{} Fix $n$ and define a labeled $n$-ary tree $\mathbb T_n$, with nodes labeled by some parameter $t$ and $n$ edges labeled $1,...,n$ emanating from each node.

To each node $t$ we associate a seed $({{\mathbf x}}, B)_t$, where ${{\mathbf x}}= (x_1,...,x_n)$ (the cluster variables) and $B$ (the mutation matrix) is a skew-symmetric $n\times n$ matrix with integer coefficients.

If the node $t$ is connected to the node $t'$ by an edge labeled by $k$, then $({{\mathbf x}}[t'],B[t'])=\mu_k({{\mathbf 