Linear isometries on Weighted Coordinates Poset Block Space

TL;DR

This paper characterizes the groups of linear isometries for weighted coordinate poset block spaces, extending known results for poset and block metrics to a more general setting involving weights and labels.

Contribution

It determines the structure of linear isometry groups for $(P,w, ext{}\pi)$-metric spaces, generalizing previous results for poset and block metric spaces.

Findings

Linear isometry groups are described as semi-direct products.

Re-derivation of isometry groups for $(P,w)$- and $(P, ext{ }\pi)$-metric spaces.

Extension of isometry characterization to weighted coordinate poset block spaces.

Abstract

Given $[n]=\{1,2,\ldots,n\}$, a poset order $\preceq$ on $[n]$, a label map $\pi : [n] \rightarrow \mathbb{N}$ defined by $\pi(i)=k_i$ with $\sum_{i=1}^{n}\pi (i) = N$, and a weight function $w$ on $\mathbb{F}_{q}$, let $\mathbb{F}_{q}^N$ be the vector space of $N$-tuples over the field $\mathbb{F}_{q}$ equipped with $(P,w,\pi)$-metric where $ \mathbb{F}_q^N $ is the direct sum of spaces $ \mathbb{F}_{q}^{k_1}, \mathbb{F}_{q}^{k_2}, \ldots, \mathbb{F}_{q}^{k_n} $. In this paper, we determine the groups of linear isometries of $(P,w,\pi)$-metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset (block) metric spaces. In particular, we re-obtain the group of linear isometries of the $(P,w)$-mertic spaces and $(P,\pi)$-mertic spaces.