= \prod_{i=0}^{|C_{{{\alpha}},\beta}|-1}
Q_{\beta,\lfloor\frac{t_\beta k+i}{t_{{\alpha}}} \rfloor},$$ where $\lfloor a \rfloor$ is the integer part of $a$. Here, $t_{{\alpha}}$ are the integers which symmetrize the Cartan matrix. Namely, $t_r=2$ for $B_r$, $t_{{\alpha}}=2$ (${{\alpha}}<r$) for $C_r$, $t_3=t_4=2$ for $F_4$ and $t_2=3$ for $G_2$, and $t_{{\alpha}}=1$ otherwise.

This is a recursion relation which can be used to express any $Q_{{{\alpha}},k}$ in terms of $2r$ initial data points. Usually, one chooses the initial conditions $\{Q_{{{\alpha}},0},
Q_{{{\alpha}},1}~|~{{\alpha}}\in I_r\}$. Then $$\label{positive}
Q_{{{\alpha}},k+1} = \frac{Q_{{{\alpha}},k}^2 - \prod_{\beta\sim {{\alpha}}}
  {\mathcal T}_k^{{{\alpha}},\beta}}{Q_{{{\alpha}},k-1}}, \ k\geq 1,$$ and $$\label{negative}
Q_{{{\alpha}},k-1}=\frac{Q_{{{\alpha}},k}^2 - \prod_{\beta\sim {{\alpha}}}
  {\mathcal T}_k^{{{\alpha}},\beta}}{Q_{{{\alpha}},k+1}}, \ k\leq0.$$

Consider Equation in the particular case of the initial conditions that for all ${{\alpha}}$, $Q_{{{\alpha}},0}=1$ and $Q_{{{\alpha}},1}={\rm Char}
W_{{{\alpha}},1}$, the character of the fundamental Kirillov-Reshetikhin module, with highest ${{\mathfrak{g}}}$-weight $\omega_{{\alpha}}$. In this case, the solutions $Q_{{{\alpha}},k}$ of are known to be the characters of KR-modules $W_{{{\alpha}},k}$ with highest ${{\mathfrak{g}}}$-weight $k
\omega_{{\alpha}}$ [@Nakajima; @Hernandez]. The recursion relation with these initial conditions is what is usually called the $Q$-system in the literature.

We do not impose these initial conditions in the definition of the cluster algebra. As we will see, this is a special (singular) point in the parameter space for this algebra. One way to see 