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.
Symmetric functions in superspace: a compendium of results and open problems (including a SageMath worksheet)
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abstract
We present a review of the most important results in the theory of symmetric functions in superspace (or symmetric superpolynomials), summarizing all principal contributions since its introduction in 2001 in the context of the supersymmetric Calogero-Moser-Sutherland integrable model. We also mention some open problems which remain unanswered at this moment, in particular the connection with representation theory. In addition, we provide a free open access source code, relying on SageMath library, that can be used as a research tool for symmetric superpolynomials. The content is directed to an audience new to this research area, but who is familiar with the classical theory of symmetric functions.
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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.