Linear Groups of Isometries with Poset Structures

Let $V$ be an $n$-dimensional vector space over a finite field $\Bbb{F}_{q}$ and $P=\{1,2,\ldots ,n\}$ a poset. We consider on $V$ the poset-metric $d_{P}$. In this paper, we give a completedescription of groups of linear isometries of the metric space $\left(V,d_{P}\right) $, for any poset-metric $d_{P}$. We show that a linear isometry induces an automorphism of order in poset $P$, andconsequently we show the existence of a pair of ordered bases of $V$ relative to which every linear isometry is represented by an $n\times n$ upper triangular matrix.