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We study certain monoidal subcategories (introduced by David Hernandez andBernard Leclerc) of finite--dimensional representations of a quantum affinealgebra of type $A$. We classify the set of prime representations in thesesubcategories and give necessary and sufficient conditions for a tensor productof two prime representations to be irreducible. In the case of a reducibletensor product we describe the prime decomposition of the simple factors. As aconsequence we prove that these subcategories are monoidal categorifications ofa cluster algebra of type $A$ with coefficients.

Tensor products and $q$-characters of HL-modules and monoidal categorifications