Similar sublattices of the root lattice A4

Abstract

Similar sublattices of the root lattice A 4 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 m 2 +mn−n 2 .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.

Similar Sublattices of Planar Lattices

“…If Γ is a planar lattice, we denote the number of distinct SSLs of Γ of index m by f (m). The integer-valued arithmetic function f is super-multiplicative, which means that one has f (mn) ≥ f (m) f (n) for coprime m, n ∈ N; see [5] and references therein for details. An example for genuine super-multiplicativity is given by the rectangular lattice 1, τ Z with τ = 3i/2; further examples will follow below.…”

“…This motivates the use of Dirichlet series as their generating functions. We thus define In previous articles, the generating functions have been calculated for a variety of examples in the plane (see [3,6] and references therein) and in higher dimensions (compare [5][6][7]10]). Standard results such as Delange's Theorem [24,Thm.…”

“…We will thus use a formulation via the (orientation preserving) similarity mappings of a lattice into itself, which form a ring in our case. Beyond the planar situation, various results are known in 3-and 4-space (via quaternions; see [4,5,7,10,27]). General results are still sparse and restricted to rather special cases; see [10,16] and references therein.…”

CSLs of the root lattice A4

“…where τ = (1 + √ 5)/2 is the golden ratio; see [2,5,6] for details. Note that L is just rescaled by a factor 1 √ 2 in comparison to A 4 .…”

Similar Submodules and Coincidence Site Modules

“…These concepts have been generalised for modules to analyse grain boundaries in quasicrystals [6,7,8,9]. On the other hand, similar sublattices and submodules have been studied [10,11,12], and it soon turned that there must be close connections between these two types of sublattices, compare for instance [12] and [13] for similar sublattices and CSLs of the A 4 -lattice. In 2008, S. Glied and M. Baake established a connection between similar sublattices and CSLs by showing that the group of coincidence isometries is a normal subgroup of the group of similarity isometries [14], a result which was later generalised to a certain class of modules [15], which the author called S-modules.…”