Fourier-Reflexive Partitions Induced by Poset Metric

Let $\mathrm{H}=\prod\nolimits_{i\in\Omega}H_{i}$ be the cartesian product of finite abelian groups $H_{i}$ indexed by a finite set $\Omega$. Any partition of H gives rise to a dual partition of its character group $\hat{\mathrm{H}}$. A given poset (i.e., partially ordered set) P on $\Omega$ gives rise to the corresponding poset metric on H, which further leads to a partition $\Gamma$ of H. We prove that if $\Gamma$ is Fourier-reflexive, then its dual partition $\hat{\Gamma}$ coincides with the partition of $\hat{\mathrm{H}}$ induced by $\overline{\mathrm{P}}$, the dual poset of P, and moreover, P is necessarily hierarchical. This result establishes a conjecture proposed by Heide Gluesing-Luerssen in [4]. We also show that with some other assumptions, $\hat{\Gamma}$ is finer than the partition of $\hat{\mathrm{H}}$ induced by $\overline{\mathrm{P}}$. We prove these results by relating the partitions with certain family of polynomials, whose basic properties are studied in a slightly more general setting.