  Let $\mathbf{H}$ be the cartesian product of a family of finite abelian
groups indexed by a finite set $\Omega$. A given poset (i.e., partially ordered
set) $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ gives rise to a poset
metric on $\mathbf{H}$, which further leads to a partition
$\mathcal{Q}(\mathbf{H},\mathbf{P})$ of $\mathbf{H}$. We prove that if
$\mathcal{Q}(\mathbf{H},\mathbf{P})$ is Fourier-reflexive, then its dual
partition $\Lambda$ coincides with the partition of $\hat{\mathbf{H}}$ induced
by $\mathbf{\overline{P}}$, the dual poset of $\mathbf{P}$, and moreover,
$\mathbf{P}$ is necessarily hierarchical. This result establishes a conjecture
proposed by Gluesing-Luerssen in \cite{4}. We also show that with some other
assumptions, $\Lambda$ is finer than the partition of $\hat{\mathbf{H}}$
induced by $\mathbf{\overline{P}}$. In addition, we give some necessary and
sufficient conditions for $\mathbf{P}$ to be hierarchical, and for the case
that $\mathbf{P}$ is hierarchical, we give an explicit criterion for
determining whether two codewords in $\hat{\mathbf{H}}$ belong to the same
block of $\Lambda$. We prove these results by relating the involved partitions
with certain family of polynomials, a generalized version of which is also
proposed and studied to generalize the aforementioned results.
