Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials

TL;DR

This paper explores the shifted quantum toroidal algebra of type _{1|1} and derives the Pieri rule for super Macdonald polynomials, connecting algebraic actions to differential operators and supersymmetric Hamiltonians.

Contribution

It introduces the shifted quantum toroidal algebra of type _{1|1} and derives the Pieri rule for super Macdonald polynomials, linking algebraic structures to differential operators and supersymmetric Hamiltonians.

Findings

Derived the Pieri rule in terms of differential operators in power sums and fermionic sums.

Expressed supersymmetric Hamiltonians via anti-commutators of super charges.

Reproduced known results on supersymmetric Hamiltonians from algebraic constructions.

Abstract

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 $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$. The action of the super charges of $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$ 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.