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