On Bravais colourings of cyclotomic integers with class number one

TL;DR

This paper characterizes the symmetry and color-preserving groups for all $ ext{n}$-colorings of cyclotomic integer modules with class number one, providing a comprehensive understanding of their structure.

Contribution

It offers a complete characterization of the colour symmetry and preserving groups for all $ ext{n}$-colourings of cyclotomic integers with class number one.

Findings

Explicit descriptions of $H$ and $K$ for all relevant $ ext{n}$-colorings.

Identification of the structure of colour groups in cyclotomic modules.

Complete classification for modules with class number one.

Abstract

Given a Bravais colouring of planar modules $\M_n:=\Z[\xi_n]$, where $\xi_n$ is a primitive $n$th root of unity, two important colour groups arise: the colour symmetry group $H$, which permutes the colours of a given colouring of $\M_n$, and the colour preserving group $K$, a normal subgroup of $H$ that fixes the colours. This paper gives a complete characterisation of $H$ and $K$ for all $\ell$-colourings of $\M_n$ for values of $n$ for which $\M_n$ has class number one.