Coincidence rotations of the root lattice $A_4$

TL;DR

This paper investigates the coincidence rotations of the root lattice A4, using its embedding into the icosian ring to derive a Dirichlet series generating function that describes their statistical properties.

Contribution

It introduces a novel approach to analyze the coincidence site lattices of A4 via its embedding into the icosian ring, enabling explicit statistical descriptions.

Findings

Derived a Dirichlet series generating function for coincidence rotations

Classified the sublattices of A4 using arithmetic properties of the icosian ring

Provided a detailed analysis of the indices of coincidence rotations

Abstract

The coincidence site lattices of the root lattice $A_4$ are considered, and the statistics of the corresponding coincidence rotations according to their indices is expressed in terms of a Dirichlet series generating function. This is possible via an embedding of $A_4$ into the icosian ring with its rich arithmetic structure, which recently (arXiv:math.MG/0702448) led to the classification of the similar sublattices of $A_4$.