Super Macdonald polynomials indexed by super partitions form a basis of the level zero super Fock module of the shifted quantum toroidal algebra U_{q,t}(gl hat hat 1|1), with the Pieri rule following from super charge actions and yielding supersymmetric Hamiltonians via anti-commutators.
Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra
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In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].
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Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials
Super Macdonald polynomials indexed by super partitions form a basis of the level zero super Fock module of the shifted quantum toroidal algebra U_{q,t}(gl hat hat 1|1), with the Pieri rule following from super charge actions and yielding supersymmetric Hamiltonians via anti-commutators.