| Paper ID | D1-S5-T2.1 |
| Paper Title |
Fourier-Reflexive Partitions Induced by Poset Metric |
| Authors |
Yang Xu, Fudan University/The University of Hong Kong, China; Haibin Kan, Fudan University, China; Guangyue Han, The University of Hong Kong, China |
| Session |
D1-S5-T2: Distance Metrics |
| Chaired Session: |
Monday, 12 July, 23:20 - 23:40 |
| Engagement Session: |
Monday, 12 July, 23:40 - 00:00 |
| Abstract |
Let $\mathbf{H}=\prod_{i\in \Omega}H_{i}$ be the cartesian product of finite abelian groups $H_{i} $ indexed by a finite set $\Omega$. Any partition of $\mathbf{H}$ gives rise to a dual partition of its character group $\hat{\mathbf{H}}$. A given poset (i.e., partially ordered set) $\mathbf{P}$ on $\Omega$ gives rise to the corresponding poset metric on $\mathbf{H}$, which further leads to a partition $\Gamma$ of $\mathbf{H}$. We prove that if $\Gamma$ is Fourier-reflexive, then its dual partition $\widehat{\Gamma}$ 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 Heide Gluesing-Luerssen in \cite{4}. We also show that with some other assumptions, $\widehat{\Gamma}$ is finer than the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$. We prove these results by relating the partitions with certain family of polynomials, whose basic properties are studied in a slightly more general setting.
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