Fourier-Reflexive Partitions and Group of Linear Isometries with Respect to Weighted Poset Metric

Let H be the cartesian product of a family of abelian groups indexed by a nonempty finite set Ω. A given poset P = (Ω, ≼
P
) and a map ω : Ω → ℝ
⁺
give rise to the (P, ω)-weight on H, which further leads to a partition $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$ of H. For the case that H is finite, we give sufficient conditions for two codewords to belong to the same block of Λ, the dual partition of $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$, and sufficient conditions for $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$ to be Fourier-reflexive. By relating the involved partitions with certain polynomials, we show that such sufficient conditions are also necessary if P is hierarchical and ω is integer valued. With H further set to be a finite vector space over a finite field $\mathbb{F}$, from a partition perspective, we extend the property of "admitting MacWilliams identity" to arbitrary pairs of partitions of H, and prove that a pair of $\mathbb{F}$-invariant partitions (Λ, Γ) with |Λ| = |Γ| admits MacWilliams identity if and only if (Λ, Γ) is a pair of mutually dual Fourier-reflexive partitions. Such a result is applied to the partition $\mathcal{Q}\left( {{\text{H}},{\text{P}},\omega } \right)$. Finally, with H set to be a (possibly infinite) left module over a ring S, we show that each (P, ω)- weight isometry of H uniquely induces an order automorphism of P, which further leads to a group homomorphism from the group of (P, ω)-weight isometries to Aut (P), whose kernel consists of isometries preserving the P-support.