Shifted quantum toroidal algebra of type ${\mathfrak{g}\mathfrak{l}}_{1\mathrm{\mid }1}$ and the Pieri rule of the super Macdonald polynomials

The super Macdonald polynomials indexed by the super partitions form a basis of the level zero super Fock module (combinatorial representation) of the quantum toroidal algebra ${\mathcal{U}}_{q,t}({\widehat{\widehat{\mathfrak{g}\mathfrak{l}}}}_{1\mathrm{\mid }1})$. The action of the super charges of ${\mathcal{U}}_{q,t}({\widehat{\widehat{\mathfrak{g}\mathfrak{l}}}}_{1\mathrm{\mid }1})$ implies the Pieri rule of the super Macdonald polynomials. We can express the Pieri rule in terms of differential operators in the power sums $p_k$ and the fermionic power sums ${\pi }_k$, which leads to the operators on the Fock space of a free boson and a free fermion. From the Pieri rule we compute the supersymmetric Hamiltonians given by the anti-commutator of the super charges and recover the results previously obtained in the literature. It is remarkable that we have to deal with a shifted quantum toroidal algebra.