$(q,t)$-deformations of multivariate hook product formulae

Abstract

We generalize multivariate hook product formulae for $P$-partitions. We use Macdonald symmetric functions to prove a $(q,t)$-deformation of Gansner's hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a $d$-complete poset, we present a conjectural $(q,t)$-deformation of Peterson--Proctor's hook product formula.