Lax operator for Macdonald symmetric functions

TL;DR

This paper constructs a family of commuting operators using the Lax operator formalism, with Macdonald symmetric functions as eigenfunctions, linking them to Hall-Littlewood functions and Baker-Akhiezer functions.

Contribution

It introduces a novel Lax operator framework for Macdonald functions, connecting them to Hall-Littlewood functions and Baker-Akhiezer functions.

Findings

Constructed commuting operators with Macdonald functions as eigenfunctions

Expressed operators in terms of Hall-Littlewood functions

Linked the formalism to Baker-Akhiezer functions

Abstract

Using the Lax operator formalism, we construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables and of two parameters $q,t$ are their eigenfunctions. We express our operators in terms of the Hall-Littlewood symmetric functions of the same variables and of the parameter $t$ corresponding to the partitions with one part only. Our expression is based on the notion of Baker-Akhiezer function.