A formula for the Jack super nabla operator
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866913060707368960 |
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| author | Dali, Houcine Ben |
| author_facet | Dali, Houcine Ben |
| contents | We study a Jack analog $\nabla(\mathbf{p},\mathbf{q})$ of the super nabla operator recently introduced by Bergeron, Haglund, Iraci and Romero for Macdonald polynomials. We prove that $\nabla(\mathbf{p},\mathbf{q})$ has a differential expression in the power-sum basis given in terms of Chapuy-Dołe\k{}ga and Nazarov-Sklyanin operators.
This result is obtained from a more general formula for the operator $G(\mathbf{p},\mathbf{q})$ encoding the structure coefficients of Jack characters, from which $\nabla(\mathbf{p},\mathbf{q})$ is obtained by taking the top homogeneous part.
A key step of the proof involves establishing that Chapuy-Dołe\k{}ga operators together with a dehomogenized version of Nazarov-Sklyanin operators have a Heisenberg algebra structure. The proof also uses a characterization of the operator $G(\mathbf{p},\mathbf{q})$ with a family of differential equations, recently established by the author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A formula for the Jack super nabla operator Dali, Houcine Ben Combinatorics 05E05 We study a Jack analog $\nabla(\mathbf{p},\mathbf{q})$ of the super nabla operator recently introduced by Bergeron, Haglund, Iraci and Romero for Macdonald polynomials. We prove that $\nabla(\mathbf{p},\mathbf{q})$ has a differential expression in the power-sum basis given in terms of Chapuy-Dołe\k{}ga and Nazarov-Sklyanin operators. This result is obtained from a more general formula for the operator $G(\mathbf{p},\mathbf{q})$ encoding the structure coefficients of Jack characters, from which $\nabla(\mathbf{p},\mathbf{q})$ is obtained by taking the top homogeneous part. A key step of the proof involves establishing that Chapuy-Dołe\k{}ga operators together with a dehomogenized version of Nazarov-Sklyanin operators have a Heisenberg algebra structure. The proof also uses a characterization of the operator $G(\mathbf{p},\mathbf{q})$ with a family of differential equations, recently established by the author. |
| title | A formula for the Jack super nabla operator |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2509.18625 |