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arXiv · 2508.19704

Generalized Macdonald functions and quantum toroidal gl(1) algebra

Jean-Emile Bourgine·Luca Cassia·Artem Stoyan
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Abstract

The Macdonald operator is known to coincide with a certain element of the quantum toroidal $\mathfrak{gl}(1)$ algebra in the Fock representation of levels $(1,0)$. A generalization of this operator to higher levels $(r,0)$ can be built using the coproduct structure, it is diagonalized by the generalized Macdonald symmetric functions, indexed by $r$-tuple partitions and depending on $r$ alphabets. In this paper, we extend to the generalized case some of the known formulas obeyed by ordinary Macdonald symmetric functions, such as the $e_1$-Pieri rule or the identity relating them to Whittaker vectors obtained by Garsia, Haiman, and Tesler. We also propose a generalization of the five-term relation, and the Fourier/Hopf pairing. In addition, we prove the factorized expression of the generalized Macdonald kernel conjectured previously by Zenkevich.

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Jean-Emile Bourgine, Luca Cassia, Artem Stoyan. 2025-08-27. Generalized Macdonald functions and quantum toroidal gl(1) algebra. https://arxiv.org/abs/2508.19704

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