Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices

Edyta Bartnicka; Metod Saniga

doi:10.1016/j.laa.2019.10.012

paper

Affine and Projective Planes Linked with Projective Lines over Certain Rings of Lower Triangular Matrices

arXiv:1903.04287 · doi:10.1016/j.laa.2019.10.012

Abstract

Let $T_{n} (q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F (q)$ of order $q \geq 2$ and let $^{2} T_{n} (q)$ denote its free left module. For $n = 2, 3$ it is shown that the projective line over $T_{n} (q)$ gives rise to a set of $(q + 1)^{(n - 1)} q^{\frac{3 ( n - 1 ) ( n - 2 )}{2}}$ affine planes of order $q$ . The points of such an affine plane are non-free cyclic submodules of $^{2} T_{n} (q)$ not contained in any non-unimodular free cyclic submodule of $^{2} T_{n} (q)$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane can be extended to the projective plane of order $q$ , with the `line at infinity' being represented by those free cyclic submodules of $^{2} T_{n} (q)$ that are generated by non-unimodular pairs. Our approach can straightforwardly be adjusted to address the case of arbitrary $n$ .