Classification of real modules in monoidal categorifications of cluster algebras

In this paper, we propose a conjectural formula for the highest $\mathrm{\ell }$-weight monomial of an arbitrary real module over a simply-laced quantum affine algebra. We verify the conjecture under a multiplicative reachability condition, answering the Hernandez–Leclerc classification problem in monoidal categorifications of cluster algebras under this condition. Moreover, we introduce the notion of cluster modules, generalizing Kirillov–Reshetikhin modules and Hernandez–Leclerc modules as special cases. We prove that cluster modules are reachable real modules, and obtain a system of equations governing $q$-characters of the prime cluster modules, providing a natural generalization of both the classical T-system relations for Kirillov–Reshetikhin modules and the exchange relations for Hernandez–Leclerc modules.