certain finite-dimensional representations of the quantum affine algebra $U_q(\widehat{{\mathfrak{g}}})$ or the Yangian $Y({{\mathfrak{g}}})$, where ${{\mathfrak{g}}}$ is a simple Lie algebra.

$Q$-systems were introduced in [@KR] for the classical algebras. They were later generalized by [@HKOTY] for the exceptional algebras and later to more complicated situations, such as twisted quantum affine algebras [@HKOTY2] and double affine algebras [@Her07].

The special modules related to $Q$-systems are called [@KunibaYsys; @KN] Kirillov-Reshetikhin modules. The fact that their characters satisfy the $Q$-system was proved by Kirillov and Reshetikhin [@KR] for ${{\mathfrak{g}}}=A_r$, by Nakajima [@Nakajima] for the simply-laced algebras and by Hernandez [@Hernandez; @Her07] in further generality.

Cluster algebras were introduced by Fomin and Zelevinsky [@FZcluster] in 2000, and are a very general algebraic tool which has since been applied in various algebraic, combinatorial and geometric contexts. In particular, they have been used to study $Y$-systems [@FZysys], which are related to $Q$-systems in the sense that both can be derived from $T$-systems [@KunibaYsys]. The $T$-systems are a consequence of the fusion relations for Yangian or quantum algebra modules.

The form of the $Q$-system suggests that it should be possible to recast it as part of a cluster algebra. The first step in this reformulation, for the 