Five-term relation and Macdonald polynomials

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.

Some mistakes are corrected and the proof is substantially expanded. Accepted in JCTA