Similar sublattices of the root lattice $A_4$

Details Similar sublattices of the root lattice $A_4$ Similar sublattices of the root lattice $A_4$

2007-02-15

arxiv.org/abs/math/0702448v2

Mathematics

Metric Geometry

17 pages, 1 figure; revised version with several additions and improvements

Scientific paper

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.

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