Quantum continuous $\mathfrak{gl}_{\infty}$: Tensor products of Fock modules and $\mathcal{W}_{n}$-characters

We construct a family of irreducible representations of the quantum continuous $gl_\infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.