  We use the dual functional realization of loop algebras to study the prime
irreducible objects in the Hernandez-Leclerc category for the quantum affine
algebra associated to $\mathfrak{sl}_{n+1}$. When the HL category is realized
as a monoidal categorification of a cluster algebra \cite{HL10,HL13}, these
representations correspond precisely to the cluster variables and the frozen
variables are minimal affinizations. For any height function, we determine the
classical decomposition of these representations with respect to the Hopf
subalgebra $\mathbf{U}_q(\mathfrak{sl}_{n+1})$ and describe the graded
multiplicities of their graded limits in terms of lattice points of convex
polytopes. Combined with \cite{BCMo15} we obtain the graded decomposition of
stable prime Demazure modules in level two integrable highest weight
representations of the corresponding affine Lie algebra.
