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arXiv · 2509.18625

A formula for the Jack super nabla operator

Houcine Ben Dali
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Abstract

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-Do\l{}e\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-Do\l{}e\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.

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Houcine Ben Dali. 2025-09-23. A formula for the Jack super nabla operator. https://arxiv.org/abs/2509.18625

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