Similar sublattices of the root lattice $A_4$
Metric Geometry
2019-07-17 v2 Combinatorics
Abstract
Similar sublattices of the root lattice are possible, according to a result of Conway, Rains and Sloane, for each index that is the square of a non-zero integer of the form . Here, we add a constructive approach, based on the arithmetic of the quaternion algebra and the existence of a particular involution of the second kind, which also provides the actual sublattices and the number of different solutions for a given index. The corresponding Dirichlet series generating function is closely related to the zeta function of the icosian ring.
Keywords
Cite
@article{arxiv.math/0702448,
title = {Similar sublattices of the root lattice $A_4$},
author = {Michael Baake and Manuela Heuer and Robert V. Moody},
journal= {arXiv preprint arXiv:math/0702448},
year = {2019}
}
Comments
17 pages, 1 figure; revised version with several additions and improvements