irreducible $U_q({{\mathfrak{g}}})$-modules is known in terms of explicit multiplicity formulas. These formulas can be derived [@KR; @HKOTY; @Hernandez; @DFK] from the $Q$-system, a recursion relation for the characters $$Q_{{{\alpha}},k}={\rm char}~ {\rm Res}^{U_q(\widehat{{{\mathfrak{g}}}})}_{U_q({{\mathfrak{g}}})}
W_{{{\alpha}},k}(\zeta),$$ in addition to a certain asymptotic property (see (C) of Theorem 8.1 of [@HKOTY]). We leave out the parameter $\zeta$ in what follows, because it only affects the action of the affine part of the algebra and not $U_q({{\mathfrak{g}}})$ or the characters $Q_{{{\alpha}},k}$.

The formulas for the decomposition coefficients and the statement that the characters of $KR$-modules satisfy the $Q$-system were both known (until they were proven) as the Kirillov-Reshetikhin conjecture.

### $Q$-systems

We introduce the $Q$-system in two steps. First, let us write down a completely general relation for a family of variables $$\{Q_{{{\alpha}},k}| ~ {{\alpha}}\in I_r, k\in {{\mathbb Z}}\}.$$ Let ${{\mathfrak{g}}}$ be a simple Lie algebra with Cartan matrix $C$. We denote the simple roots ${{\alpha}}$ by the corresponding integers in $I_r=\{1,...,r\}$. The $Q$-system associated with ${{\mathfrak{g}}}$ is a recursion relation of the form $$\label{Qsystem}
  Q_{{{\alpha}},k+1} Q_{{{\alpha}},k-1} = Q_{{{\alpha}},k}^2 - \prod_{\beta\sim{{\alpha}}}
  {\mathcal T}^{({{\alpha}},\beta)}_k, ~{{\alpha}}\in I_r, ~k\in {{\mathbb Z}},$$ where ${{\alpha}}\sim\beta$ means that ${{\alpha}}$ is connected to $\beta$ in the Dynkin diagram, and $$\label{T}
{\mathcal T}^{({{\alpha}},\beta)}_k 