Coincidence indices of sublattices and coincidences of colorings

TL;DR

This paper explores how the coincidence indices of a lattice and its sublattices relate through colorings, introducing the concept of color coincidences and providing examples with cubic lattices.

Contribution

It generalizes color symmetries to color coincidences, linking coincidence indices of lattices and sublattices via coloring methods.

Findings

Coincidence indices can differ between a lattice and its sublattice.

Colorings induced by sublattices reveal relationships between coincidence indices.

Examples with cubic and hypercubic lattices illustrate the concepts.

Abstract

Even though a lattice and its sublattices have the same group of coincidence isometries, the coincidence index of a coincidence isometry with respect to a lattice $\Lambda_1$ and to a sublattice $\Lambda_2$ may differ. Here, we examine the coloring of $\Lambda_1$ induced by $\Lambda_2$ to identify how the coincidence indices with respect to $\Lambda_1$ and to $\Lambda_2$ are related. This leads to a generalization of the notion of color symmetries of lattices to what we call color coincidences of lattices. Examples involving the cubic and hypercubic lattices are given to illustrate these ideas.