Generalized Macdonald functions and quantum toroidal gl(1) algebra

TL;DR

This paper extends Macdonald symmetric functions to higher levels within the quantum toroidal gl(1) algebra, providing new formulas, identities, and a proof of a conjectured kernel expression, advancing the algebraic understanding of these functions.

Contribution

It introduces a higher-level generalization of Macdonald functions, extending known formulas and proving a key conjecture about the kernel expression.

Findings

Extended Macdonald functions to higher levels using coproduct structure.

Derived generalized Pieri rules and identities for these functions.

Proved the factorized form of the generalized Macdonald kernel.

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.