Quantum continuous $g{l}_{\mathrm{\infty }}$: Tensor products of Fock modules and $W_n$ characters

We construct a family of irreducible representations of the quantum continuous $g{l}_{\mathrm{\infty }}$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $g{l}_n$ 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.