  The infinite affine Lie algebras of type ABCD, also called
$\widehat{\mathfrak{gl}}(\infty)$, $\widehat{\mathfrak{o}}(\infty)$,
$\widehat{\mathfrak{sp}}(\infty)$, are equivalent to subalgebras of the quantum
$W_{1+\infty}$ algebras. They have well-known representations on the Fock space
of either a Dirac fermion ($\hat A_\infty$), a Majorana fermion ($\hat
B_\infty$ and $\hat D_\infty$) or a symplectic boson ($\hat C_\infty$).
Explicit formulas for the action of the quantum $W_{1+\infty}$ subalgebras on
the Fock states are proposed for each representation. These formulas are the
equivalent of the \textit{vertical presentation} of the quantum toroidal
$\mathfrak{gl}(1)$ algebra Fock representation. They provide an alternative to
the fermionic and bosonic expressions of the \textit{horizontal presentation}.
Furthermore, these algebras are known to have a deep connection with symmetric
polynomials. The action of the quantum $W_{1+\infty}$ generators leads to the
derivation of Pieri-like rules and q-difference equations for these
polynomials. In the specific case of $\hat B_\infty$, a q-difference equation
is obtained for $Q$-Schur polynomials indexed by strict partitions.
