case where ${{\mathfrak{g}}}$ is simply-laced, was derived in [@Ke07]. In the current article, we give a generalization of this case to non-simply laced ${{\mathfrak{g}}}$. In this formulation, the $Q$-system appears in a very simple and easily generalizable form. We note that it does not appear to be directly related to the cluster algebra coming from the $Y$-system studied by Fomin and Zelevinsky [@FZysys].

In [@DFK], we proved a combinatorial identity (“the $M=N$” conjecture of [@HKOTY]) which implies the proof of the combinatorial Kirillov-Reshetikhin conjecture. In our proof, we introduced what we called the deformed $Q$-system, depending on an increasing number of formal variables. A specialization of this system can be expressed as the $Q$-system with general boundary conditions (not corresponding to Kirillov-Reshetikhin characters), or as a cluster algebra. It is essential for the formulation as a cluster algebra to have this more general system.

Our proof of the $M=N$ identity depends crucially on the fact that the KR-characters are polynomials in the fundamental KR-characters. This can be rephrased in terms of the cluster algebra, by considering a specialization of the initial cluster variables to the special point which gives the KR-characters at the other nodes of the cluster graph. We 