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

TL;DR

This paper introduces a $(q,t)$-deformation of multivariate hook product formulas for $P$-partitions using Macdonald symmetric functions, extending known results and proposing new conjectures for $d$-complete posets.

Contribution

It generalizes existing hook product formulas to include $(q,t)$-deformations and presents a conjecture for $d$-complete posets.

Findings

Proved a $(q,t)$-deformation of Gansner's hook product formula.

Extended the unshifted case results by Adachi.

Proposed a conjectural $(q,t)$-deformation for $d$-complete posets.

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.