Five-term relation and Macdonald polynomials

TL;DR

This paper proves a five-term relation for operators on symmetric functions, establishing a key result that confirms a conjecture by Bergeron and Haiman related to Macdonald polynomials.

Contribution

It demonstrates a five-term relation for specific operators, providing a proof for the generalized recursion conjecture in the context of Macdonald polynomials.

Findings

The five-term relation holds for certain operators on symmetric functions.

The generalized recursion conjecture of Bergeron and Haiman is confirmed.

The relation advances understanding of Macdonald polynomial recursions.

Abstract

The non-commutative five-term relation $T_{1,0} T_{0,1} = T_{0,1} T_{1,1} T_{1,0}$ is shown to hold for certain operators acting on symmetric functions. The "generalized recursion" conjecture of Bergeron and Haiman is a corollary of this result.