Fourier-Reflexive Partitions Induced by Poset Metric

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 Gluesing-Luerssen, 2015. 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.