Similar Submodules and Coincidence Site Modules

Zeiner, Peter

doi:10.12693/aphyspola.126.641

Cited by 3 publications

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“…They do not form a group, nor even a semigroup, as Scal Γ (R) Scal Γ (S) ⊆ Scal Γ (RS) is usually a proper inclusion. Nevertheless, we can still extract some information on the denominator from this inclusion; compare [97,98]. As den Γ (R) den Γ (S) must be in scal Γ (RS), it is an integer multiple of den Γ (RS), that is,…”

Section: Similar Sublatticesmentioning

confidence: 99%

“…Let us now see how this observation can be put on a more general basis. We will formulate the main results immediately for modules; compare [97,98]. For the special cases of lattices, we refer to [45].…”

Section: 5mentioning

confidence: 99%

“…The corresponding results for a special class of modules, namely the S-lattices from Definition 6.2, can be found in [44]. It is often helpful to know some connections between the coincidence indices and the corresponding denominators; compare [97].…”

Section: 5mentioning

confidence: 99%

“…Actually, some years ago, a close connection between similar sublattices (SSLs) and CSLs has been established [45], which was later generalised to Z-modules embedded in R d [44,97]. This provides the link for our two main topics, namely the enumeration of coincidence site lattices and similar sublattices, and its generalisation to embedded Z-modules.…”

mentioning

confidence: 93%

“…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 close connection between similar sublattices (SSLs) and CSLs has been established [45], which was later generalised to Z-modules embedded in R d [44,97]. This provides the link for our two main topics, namely the enumeration of coincidence site lattices and similar sublattices, and its generalisation to embedded Z-modules.Let us give an outline of this chapter.…”

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confidence: 99%

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Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Baake¹,

Zeiner²

Aperiodic Order

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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...

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Section: Similar Sublatticesmentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

mentioning

confidence: 93%

mentioning

confidence: 99%

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Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Baake¹,

Zeiner²

Aperiodic Order

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Similarity isometries of point packings

Arias

Loquias

2020

Acta Cryst Sect A

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A linear isometry R of {\bb R}^{d} is called a similarity isometry of a lattice \Gamma\subseteq{\bb R}^{d} if there exists a positive real number β such that βRΓ is a sublattice of (finite index in) Γ. The set βRΓ is referred to as a similar sublattice of Γ. A (crystallographic) point packing generated by a lattice Γ is a union of Γ with finitely many shifted copies of Γ. In this study, the notion of similarity isometries is extended to point packings. A characterization for the similarity isometries of point packings is provided and the corresponding similar subpackings are identified. Planar examples are discussed, namely the 1 × 2 rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, similarity isometries of point packings about points different from the origin are considered by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing are computed and compared with those of the hexagonal packing.

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Structure of coincidence isometry groups

Deng

Zhao

2021

Open Mathematics

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Let L L be a lattice of rank n n in an n n -dimensional Euclidean space. We show that the coincidence isometry group of L L is generated by coincidence reflections if and only if L L contains an orthogonal subset of order n n .

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Similar Submodules and Coincidence Site Modules

Cited by 3 publications

References 18 publications

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Geometric Enumeration Problems for Lattices and Embedded ℤ-Modules

Similarity isometries of point packings

Structure of coincidence isometry groups

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