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Let $\mathbf{H}$ be the cartesian product of a family of finite abeliangroups indexed by a finite set $\Omega$. A given poset (i.e., partially orderedset) $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ gives rise to a posetmetric 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 dualpartition $\Lambda$ coincides with the partition of $\hat{\mathbf{H}}$ inducedby $\mathbf{\overline{P}}$, the dual poset of $\mathbf{P}$, and moreover,$\mathbf{P}$ is necessarily hierarchical. This result establishes a conjectureproposed by Gluesing-Luerssen in \cite{4}. We also show that with some otherassumptions, $\Lambda$ is finer than the partition of $\hat{\mathbf{H}}$induced by $\mathbf{\overline{P}}$. In addition, we give some necessary andsufficient conditions for $\mathbf{P}$ to be hierarchical, and for the casethat $\mathbf{P}$ is hierarchical, we give an explicit criterion fordetermining whether two codewords in $\hat{\mathbf{H}}$ belong to the sameblock of $\Lambda$. We prove these results by relating the involved partitionswith certain family of polynomials, a generalized version of which is alsoproposed and studied to generalize the aforementioned results.

Fourier-Reflexive Partitions Induced by Poset Metric