Similar sublattices of the root lattice A4

Baake, Michael; Heuer, Manuela and Moody, Robert V. (2008). Similar sublattices of the root lattice A4. Journal of Algebra, 320(4) pp. 1391–1408.

DOI: https://doi.org/10.1016/j.jalgebra.2008.04.021

Abstract

Similar sublattices of the root lattice A4 are possible [J.H. Conway, E.M. Rains, N.J.A. Sloane, On the existence of similar sublattices, Can. J. Math. 51 (1999) 1300–1306] for each index that is the square of a non-zero integer of the form m²+mn−n². Here, we add a constructive approach, based on the arithmetic of the quaternion algebra H(Q(√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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