Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Abstract

In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to Z-modules embedded in R d . R d , which led to the notion of coincidence site modules (CSMs). They are used to describe grain boundaries in quasicrystals; compare [15,72,88] and references therein.This new development also triggered a more detailed study of lattices in dimensions d > 3, as they are used to generate aperiodic point sets by the now common cut and project technique; compare [9, Ch. 7]. In particular, lattices in dimension d = 4 such as the hypercubic lattices [4,94] and the A 4 -lattice [11,56] were studied.Further applications of CSLs can be found in coding theory in connection with so-called lattice quantisers, where lattices in large dimensions and with high packing densities are important; compare [34,85] for general background, as well as [1] for concrete applications of the A 4 -lattice and [2] for the hexagonal lattice. However, not much is known about lattices in dimensions d > 5, although there are some partial results for rational lattices [99,100,57].The original concept of CSLs has been generalised in several ways. In particular, one may study the intersection of several rotated copies of a lattice, which are known as multiple CSLs; compare [8,95,18]. They have applications to so-called multiple junctions [40,41,42], which are multiple crystal grains meeting at some common manifold. Whereas classical CSLs involve only linear isometries, one may consider affine isometries as well, which is directly related to the question of coincidences of crystallographic point packings; compare [64,66,63]. The latter are connected to the problem of coincidences of coloured lattices and colour coincidences [65,63,67].The planar case is certainly the best studied. Here, also a connection between CSLs and well-rounded sublattices has been established [17]. Moreover, even some results for the hyperbolic plane [78] have been found.Naturally, CSLs are not the only sublattices that are of interest in crystallography and coding theory. Classifying sublattices with certain symmetry constraints has a long tradition in mathematics and in crystallography; compare [79,80] and references therein. An interesting question is the number of sublattices that are similar to its parent lattice. It has been answered in detail for a considerable collection of lattices [7,14,12] in dimensions d ≤ 4. For higher dimensions, some existence results have been obtained by Conway, Rains and Sloane, who were motivated by problems in coding theory [26].Actually, some years ago, a cl...

Coincidences in four dimensions

“…Since any (positive) integer can be written as a sum of four squares, this implies that the spectrum σ is the set of all odd natural numbers. The CSLs can be calculated explicitly [11]. They read…”

“…Similarly, one can calculate the number f D4 (n) of different CSLs of a given index n. Clearly, f D4 (n) ≤ f rot D4 (n), and we do not have equality in general, since two CLSs that are not symmetrically equivalent may generate the same CSL. In fact, we have the following theorem which tells us when two CSLs are equal [11].…”

“…Along the same lines as for the hypercubic lattices one can calculate the number of coincidence rotations and the number of CSLs of a given index n. The expressions obtained are slightly more complicated, since one has to distinguish between the three cases p = 5, p = ±1 (mod 5) and p = ±2 (mod 5), as these primes behave differently and correspond to the ramifying, splitting and inert primes of Z[τ ]. If 120f rot A4 (n) is the number of coincidence rotations -recall that the point group of L contains 120 elements -its generating Dirichlet series is given by Similarly, we have for the number of CSLs Here, a criterion analogous to Theorem 2.1 exists, which will be published elsewhere [11].…”

“…If 7200f rot I (n) denotes the number of coincidence rotations and f I (n) the number of CSMs, we get again f rot I (n) ≥ f I (n). Again, equality fails to hold in general, and a complete analogue of Theorem 2.1 tells us which CSMs are equal [11].…”

Coincidence Site Modules in 3-Space

“…Quaternions are certainly also helpful in 4-space, and further progress in this direction is in sight [7], at least for simple coincidences. More complicated, however, seems the situation in higher dimensions, even for the class of root lattices, and that might be a good problem to solve.…”