Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866913134624636928 |
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| author | Kanno, Hiroaki Ohkawa, Ryo Shiraishi, Jun'ichi |
| author_facet | Kanno, Hiroaki Ohkawa, Ryo Shiraishi, Jun'ichi |
| contents | 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 $π_k$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16773 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials Kanno, Hiroaki Ohkawa, Ryo Shiraishi, Jun'ichi Quantum Algebra High Energy Physics - Theory Mathematical Physics 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 $π_k$, 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. |
| title | Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials |
| topic | Quantum Algebra High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2605.16773 |