Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

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.