Demazure modules of level two and prime representations of quantum affine $\mathfrak{sl}_{n+1}$

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to $\mathfrak{sl}_{n+1}[t]$--stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. More generally, we prove that any level two Demazure module is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.