Similar sublattices of the root lattice \(A_4\)

Similar sublattices of the root lattice \(A_4\) 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 \(m^2 + mn - n^2\). Here, we add a constructive approach, based on the arithmetic of the quaternion algebra \(\mathbb{H...

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Bibliographic Details
Published in:arXiv.org 2008-06
Main Authors: Baake, Michael, Heuer, Manuela, Moody, Robert V
Format: Article
Language:English
Subjects:
Dirichlet problem
Lattices (mathematics)
Quaternions
Subgroups
Online Access:Get full text
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Summary:Similar sublattices of the root lattice \(A_4\) 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 \(m^2 + mn - n^2\). Here, we add a constructive approach, based on the arithmetic of the quaternion algebra \(\mathbb{H} (\mathbb{Q} (\sqrt{5}))\) 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.
ISSN:2331-8422